Good morning everyone, it's great to see such a full room. It seems only like a few weeks ago that I was talking with many of you about the president's trip to Europe. And in fact we now return to Europe again as the president was there just in June where he traveled to Warsaw, Brussels and Normandy. I'm going to begin, I'm going to sort of do the trip in order so we'll begin with the stop in Tallinn, Estonia. And then we'll move on to the NATO summit discussion. I'll do sort of an overview and give you some of the context of that visit. And then Kath will give you much more of the details and the defense capabilities and can also talk about obviously the global complexities today. We sort of are going to span the world a little bit beyond the president's trip to Europe. Beginning in Tallinn, this is the second time an American president has traveled to Tallinn. The very first time was President George W. Bush in November of 2006 prior to a NATO summit that was being held in Riga at that time. Although President Obama just saw Estonian President Thomas Ilvis just last August, he met with the three Baltic presidents in the Oval Office in August. He spoke with them by phone in March just prior to Russia's annexation of Crimea. President Biden met with the Baltic leaders in March so as you can tell, the crisis over Ukraine has certainly brought to the fore a deep engagement between President Obama and the Baltic leadership. I can't begin to tell you how critical it is that the president visit NATO's new front line. And I found it was interesting as I was preparing for this discussion that in March, as the president was discussing with Baltic leaders, Vice President Biden was visiting, they talked about America's unwavering commitment to the Baltic states. If you noticed last week's NSC press release regarding this trip, they've now used the word ironclad commitment. I don't know if you've seen ironclad very often in presidential statements, I haven't. And I think it's important to highlight, and the quote was, this trip is a chance to reaffirm our ironclad commitment to Article 5 as the foundation of NATO and why ironclad. There's always been a great fear in the Baltic states that if push came to shove, they questioned whether NATO would really have their back. And I think it's very clear, not only with words and our solidarity, but we've actually put U.S. soldiers hardware in the Baltic states. I had the opportunity to be in Tallinn in April, participating in a conference, and it was the day before U.S. soldiers were due to arrive in Estonia. I can tell you my own impression was it was like the Estonians just let out their breath. They came, and in fact that was the most meaningful and significant event that U.S. soldiers had arrived in Estonia. So this is putting this into context. This is why the President's visit to Tallinn is so important, not only to provide our continued reassurance to speak to the Baltic nations to get their feedback on the current situation, but to be a physical presence in the Baltic states. And I'm sure they're looking forward to hearing President Obama's thoughts on America's future military presence in the Baltic states, and particularly potentially pre-positioning equipment, and I'll talk about that in a second about NATO. Again, just to again give you historical context, just on Saturday was the 75th anniversary of the Molotov-Ribbentrop pact. That was the secret pact between Hitler and Tallinn that carved up Central Europe. And I note that anniversary because it was also the 25th anniversary on Saturday of what is called the Baltic Way. And that is when, on August 23rd, 1989, you had literally hands across the Baltic states where over 2 million Latvians, Estonians and Lithuanians held hands across a 600 kilometer measure across the Baltic states, and it was to seek independence for fight for freedom. This historical memory here is so very fresh, and I want you to have that context when you go to, if you're traveling with the president, to Estonia. And this is why events in Ukraine, the annexation of Crimea, this is exactly why this is so palatable. The Baltic states feel that history is repeating itself, and that they want to be sure that the history of the 20th century does not become the history of the 21st century. So it is within that context that the president will arrive. We know he'll meet with the Estonian president, Prime Minister, he'll meet with the three Baltic states. I'm sure the White House will have more details.  We'll keep our fingers crossed. It would be great to hear a very important presidential message about the way forward in the region, and certainly hearing perhaps he will visit with the U.S. forces that are currently stationed in Estonia. So we'll see how that trip works out, but it'll be incredibly important. And I think, again, as a final note to remember, Estonia is a dramatic success story. As a former Soviet state, you are now looking at one of the most modern European economies that puts e-governance and e-forms to shame, flat tax, dynamic. It meets all its commitments, both to the EU and NATO. And it's an outstanding example. So I think you will enjoy, if you are on the trip, you will enjoy Tallinn very much. So let me very briefly segue to the president's next stop, which is Cardiff and Wales. And again, the main theme will be NATO's unified message vis-a-vis Russia and Ukraine. The last time a NATO summit was held was here in the United States in Chicago in 2012, and that summit was actually quite overshadowed by the Euro crisis. And of course, this summit is in fact quite overshadowed not only by Ukraine, but obviously events as they're unfolding in Iraq, Syria, and Libya. This is Secretary General Anders Foghrasmusen's last summit as Secretary General, and NATO will be welcoming former Norwegian Prime Minister Jens Stoltenberg as the new Secretary General. In some ways, the summit represents the last summit on Afghanistan, something a subject matter that NATO has been seized with for over a decade. There is obviously great uncertainty about NATO's post-ISAF footprint, resolute support, and obviously that weighs very heavily on the question mark of whether there will be a bilateral support agreement between the United States and Afghanistan and what NATO's assurances will be in the individual countries as they leave forces behind for training purposes. This unfortunately will not be a summit about enlargement, as the Chicago summit in 2012 was also not a summit about enlargement. At this point, again, within the NATO alliance, there is not consensus about opening the door. There will be encouraging words. The open door policy certainly remains there, but in fact, there is just right now no appetite, political appetite to enlarge NATO. But I think the newsmaker here at the summit will be about NATO's collective defense response in central Europe, in northern Europe, in the Baltics, and it will focus on NATO's readiness action plan. This, as it will be announced, and I know Secretary General Russ Musin has given some interviews outlining this, this is basically going to be NATO's response to the Ukraine crisis, but now starting to move into a long-term thought. This is not an issue within the alliance, that there is total agreement. Many have sought, and certainly the Baltic states in Poland have sought, a permanent NATO presence in their countries. Some allies, particularly the Germans, are very concerned that this runs against the 1997 NATO Russia founding act that the alliance would not place permanent facilities in central Europe or in the new members, but they're no longer new members after 15 years. This package, we think, will include an enhanced presence in Poland, incision, where NATO has a multinational corps northeast. Right now the Germans, the Danes, have taken a leadership role in developing that, and I think you'll see where the potential of prepositioning equipment, it will not be permanent, it will be rotating, and it will be called a persistent presence, rather than a permanent presence. Again, I think these are word choices to allow greater comfort within the alliance, but quite frankly this will be a permanent rotating presence for the foreseeable future, and I think that's the best way to think about it. There will be lots of discussion about individual NATO members increasing their defense spending. We have said this for the past 15 years at every NATO summit that has ever been discussed, but actually members will start suggesting that they're meeting those targets. Poland is putting forward a very robust military modernization program. The Baltics are trying to increase, Estonia should be noted, will be a 2% defense spending per GDP, but we need our allies to do much, much more. Finally, on Ukraine, President Poroshenko will be at the summit, he'll be the only invited leader. There will be a NATO Ukraine summit, and what we're hearing, that there will be announcements about some trust funds for Ukraine that will support the Ukrainian military logistics, command and control, cyber defense, and trying to help defray costs to support the military. We'll see how details come forward on that. I'll segue this to Kath. Obviously there will be a discussion at the summit on events as they're unfolding in Iraq, in Syria, allied support for the Kurds, and I'm sure, and I'll let Kath really focus on that. There will be, although it will not be the main focus, there will be discussion, I'm sure, on the margins and the sidelines of allied support in these emerging operations. Last note, and I'm sorry that I've gone on for so long. I want to give you a little context. As the President arrives in Wales, it comes at a very historic moment, exactly two weeks before the Scottish referendum. Let's just say we've been watching some very lively debates as recently as Monday about the referendum. Right now, if you believe polls, there's a 10% gap between the yes and the no votes with a no to independence at about 57%. Yes votes are 43%. The President actually addressed this issue in June after his bilateral meeting with David Cameron, and I thought the formula was about right. The President said that the United States has a deep interest in making sure that one of our closest allies we will ever have remains a strong, robust, united, and effective partner. I would say that puts us in the better together column, but he hastened to add it is for the Scottish people to decide. I am sure that formulation will remain exactly the same as he has asked many, many questions about that when he arrives in Wales. Just as a side note, Congress has weighed in on this as well. There was a House resolution that basically said that we support a united, secure, and prosperous United Kingdom as an essential for U.S. national security priorities in Europe. So bipartisan support for a united kingdom. Clearly, huge implications for NATO. Should the Scottish people decide to vote for independence? Very big questions about British nuclear deterrence that is based in Scotland. Whether Scotland would seek to become a NATO member. So maybe that enlargement conversation would reopen. So all of this is a very dramatic backdrop where Europe's security environment has turned upside down. NATO had never thought nine months ago that it would be focusing on a robust collective defense posture in Northern Europe, but that's exactly where we are today. And so with that, I'll handle the rest of the world. Good morning. I'm Kathleen Hicks. I direct the international security program here. Let me first sweep up just a few things, much less of a strategic look, but just a few things also going on at the summit that I just want to highlight. Just touching on Afghanistan, obviously the big question of Afghanistan is who will the Afghan government send, if anyone, to the summit. There was in 2012 this opportunity for the Afghans to participate. President Karzai came at that time. We obviously have a contested election right now in Afghanistan. As a matter of fact, in the last 24 hours, we've had the threat of one candidate potentially polling out of that contested election. So there's a lot of turmoil in Afghanistan, and for the NATO summit, it's now becoming quite a crisis, if you will, of who will come and represent the Afghan government. Will it be the two potential next presidents? President Karzai has said it won't be him, and given that he does not support the BSA being signed is the reason he's given, and then there could be a different representative from the Afghan government. But I think making sure someone from the Afghan government comes will be an important signal for NATO that it is continuing its commitment to Afghanistan and it's not a rear-view mirror issue. Another item to note that's on the summit agenda is that the NATO charter isn't going to be updated. The intention is for the NATO to update its charter to include cyber security as something to be covered under Article 5. I suspect the language will be very vague, so it will not be immediately clear what that will mean in terms of what type of response NATO might have in response to any kind of cyber attack. But it will be notable, I think, that cyber will now be explicitly called out as covered under Article 5, which is the collective defense article of the NATO charter. A couple other things on the readiness action plan that Heather spoke to. I do think the terminology that they are using, Rasmussen and General Breedlove, is this fitter, faster, and more flexible. Obviously they like a literation approach, and it is hard to say. A strategy for the future. I think the way that Heather's laid out is exactly right. It's sort of not even a warm-basing, but a hot-basing approach where they'll build off of this Polish facility that NATO already has on the Baltic very close to the German border, convenient to the Danes. And that's a place where we will begin as an alliance to pre-position more equipment to have forces flow through quite routinely. And the big question on the U.S. side, across NATO, but on the U.S. side, is how the U.S. will support that, what the strain on our own forces is around the world, and how much commitment the U.S. will put forward into Eastern Europe. If there's anything that the Baltics do trust within NATO, it's a U.S. commitment directly. So they will be pushing hard that that NATO contingent will have a heavy U.S. signal in it. That's what they trust most. If there's a U.S. component there, that there's skin in the game, so to speak, in terms of defensive activity. Another initiative coming out of this summit is what is being called the Defense Capacity Building Initiative, and this relates to the point that there will not really be an enlargement discussion per se, but there is an emphasis given the Ukraine crisis on how NATO will work with both Eastern and Central European countries, and then those out-of-area partners. Recall that NATO has partners like Brazil and others that it is trying to work with, Colombia. And so one of the initiatives going forward into this summit will be how do we as an alliance help other nations build effective defenses, and parenthetically while not allowing them into the alliance. So it will be important to see what kind of promises or commitments NATO makes in terms of advise and assist support to these other countries, particularly those in Central and Eastern Europe who will not be put forward for membership. And then also, as Heather mentioned, the 2% comes up every 2% commitment, the standard CSIS draft work that we have to date shows only four countries are at that level right now, US, UK, Greece, and Estonia. The question has always been whether that benchmark matters anymore, whether it makes sense. The Greeks, for instance, meet the 2%, but I think you'd be hard-pressed to point to a lot of high-quality Greek capability that we're relying on in NATO. So there will be, as there always is, this conversation around what is the right measure to determine how effective allies are in their commitments to NATO. I don't think you'll see an incredible advancement in that debate, but it will continue, and I think the 2% benchmark is increasingly irrelevant and will become even more clearly so in the initiatives that roll out. Moving on beyond Europe, the crisis in Iraq, obviously, is in Syria and elsewhere in the Middle East, Gaza, and Israel. Those are very much on the minds of the NATO members, but Rasmussen is working hard to keep it off of the formal agenda. I think he will succeed there. I do think you'll see something written in the communiques and any public statements coming out of leaders, certainly in the bilateral discussions that happen on the sideline of the summit. But NATO itself is trying to ensure that it is staying out as an alliance of any kind of military intervention with regard to the crisis in Iraq. I think you will most certainly see a recommitment, a restatement of the standing commitment to defense of Turkey as a NATO ally should it feel threatened. And as you may recall, Turkey has called for Article 4 consultations before with regard to the Syrian crisis. So I think there'll be an effort to assure the Turks and even others along the southern borders of NATO that the alliance is standing by those members. Nevertheless there will have to be an acknowledgement that individual European countries and certainly the United States are taking action, actively working militarily on issues relating to Iraq and then potentially Syria. We'll see how the week goes. Those I think are the biggest outside issues. There has been some discussion about the Asia rebalance before, certainly before the Ukraine crisis and how the Asia rebalance on the U.S. side should affect NATO, should NATO rebalance. Again, I think while that at one point may have been a major discussion for this summit, I think it will fall much lower in the agenda of those sidebar conversations because of the press business with regard to Afghanistan, certainly the Russia and Eastern Europe elements and now the Middle East. So let me stop there and we're open to questions. Great, we're going to open up to questions. I think ourselves will be helpful for the transcript. Let's start with Julie. Hi, I'm Julie Pace from AP. I had a question on Afghanistan. It seems like from the U.S. perspective the Obama administration is giving the Afghans a lot of leeway to try to figure out their politics before saying they would have to pull all the troops out. And I'm wondering what their perspective is from NATO members, both the alliance as a whole and also some of the individual countries that could potentially keep a commitment after this year. You know, I think there is no one NATO. So let me break it down to maybe a U.K., Western European view and the U.S. view. I do think you've accurately captured that. I think we are being respectful, perhaps is the way to put it, to the Afghan political process. I think there's a lot of nervousness under that and particularly on the military side in terms of the timelines associated with if there's not a BSA for the U.S. and Ergo, not a BSA for NATO with Afghanistan, that there would have to be a very stressful timeline to pull out troops. But I think there is this because both of the candidates have been committed to the BSA. We're playing a little bit of a game of chicken here. And I think bottom line is you'll continue to get this sense from those who are looking to maintain commitment, including the United States, of we respect the Afghan political process mostly because we believe it will end up with a BSA and a nervousness that should something happen there that there would have to be a relatively rapid drawdown. I think the larger question really is about the hangover, if you will, of the shadow. Maybe it's a better way to put it, of what's happening in Iraq right now and the sense of whether that should affect, you know, change the calculus with regard to the timeline commitment in Afghanistan. I think there it's much less clear right now that the United States and certainly NATO allies, as best we can tell, are holding the line in terms of their decisions to ramp down on a timetable. But there is also time in that timetable for a change of position on that should the Afghan political process stall or other changes in the Afghan security environment cause a change of approach from NATO to be warranted. Just to finger that, I think in some ways the European allies have been just in a constant wait mode. First they had to wait until the administration announced how many forces it was leaving and that was going to be weighing on its decisions, particularly on force protection issues of the trainers. Now they're waiting for this and they will then wait for the United States to see what happens. So they're the last in the chain in some ways to respond and they have to see how all of these things shift out. The Germans and the Italians have already put forward what they would provide to Resolute Support but again I think that is completely contingent even that commitment on how all of the sequential problems happen and again I think what I'm hearing from allies there is a concern on force protection for these trainers particularly in the north how far are they going to be able to be and what support are they going to have. There's just a lot of questions but they feel like they don't have to act right now they have to wait until all of this works its way through. Mr. Condon. Thanks. Two questions. Heather you mentioned that the European security environment has been turned upside down. Can you talk a little more about how frustrated they are that they can't ever follow the agenda that they set. They're always reacting to things like they were in Chicago. And secondly President Obama his leadership is very much under fire domestically. Is there a particular challenge for him at this summit? Is there something that the allies are looking for that we should be looking for? Yes it's every person's ever worked in government you have the most well planned agenda perfectly scripted thing and then events my dear boy events as Harold McMillan would say always change the calculation. Turning upside down you know quite frankly the predictions no one predicted this no one predicted that a portion of Europe would be annexed in 2014 that we would basically we would have a war between Russia and Ukraine and I think it's time we begin to state very clearly what this is and it is now required NATO to really adapt and change fairly dramatically in some ways NATO should thank Vladimir Putin because it was really searching for its purpose post ISAF and it was having a fairly significant identity crisis as people were looking towards the summit 9-12 months ago and it has now not only been repurposed it's been reinvigorated. And you're seeing some fairly extraordinary turn of events again with this pre-positioning of equipment I think a persistent presence by NATO in the Baltic states in Poland for the foreseeable future that was never envisioned and would never have been part of the calculation were it not for Vladimir Putin's actions so that has been cleared. Now the challenge here is that not all of the alliance feels the same way and that's what we're dealing with this is palatable in the Baltic states and in Poland I would add Romania to that as well but it's certainly not palatable in other central European countries like Hungary that do not believe there's any issue and in fact are criticizing that Europe is sanctioning Vladimir Putin so we have unevenness and you also have unfortunately the French government continuing its sale of the two Mistral ships so you have while unity and you'll see here strong, strong messages of solidarity at NATO and unity of purpose there are still some very clear discordant notes about what this threat is long term and let me just say we have to begin to start focusing on what the long term policy is here we are dealing with the hour by hour what is the action, what is the humanitarian convoy we're dealing with the moment by moment and we need to start as a transatlantic community putting together a new policy formulation because this crisis is not going to weigh if anything it's going to be a long term point of instability on leadership it's a great question yes NATO as it has historically we'll be looking for strong US leadership here and as Kath mentioned while this is an alliance response at its core it must be the United States that's the ultimate security guarantee in the minds of many that is the article 5 commitment that's why my Estonian friends sort of let that breath out that they were holding in for many weeks but we cannot do it alone and we do now need the rest of the alliance to step forward and on leadership I think what we've seen through this crisis has been the fact that German Chancellor Angela Merkel has actually come to the fore as a critical leadership voice in Europe towards Russia it's a very complicated voice in policy because of German domestic politics and just this is a dramatic change of events that no one had anticipated but I think that the president does have another very strong leader that can help him propel this policy and I think that should be noted as well let's go to Michael very briefly then Kath please weigh in I think in many ways the individual European countries are wrestling with this obviously it has been a very dramatic debate within the UK about the foreign fighter problem but we've also seen this concern in Germany, France and elsewhere I think Mike the challenge is being addressed through a lot of intelligence channels bilateral channels that's not a discussion that's going to be a NATO discussion per se I am sure again in the bilateral meetings a lot of conversations are going to be taken place that just happens in different channels but you're absolutely right the threat from within Europe can be as significant as an external challenge no I think that's right I mean if I had to gas on the rationale there I think they do have a strong agenda that they want to push through that's Russia focused that's focused on NATO, Quad NATO if you will and I do think that they want to drive that it's been very clear through Rasmussen's public statements the op-eds he's put out even Cameron's pieces that have come out they are driving toward making this summit matter it's his last summit he wants to leave it with a strong sense of accomplishment vis-a-vis the Russia-Ukraine issue and how it affects Eastern Europe so I honestly think it's as simple as that it's also not a terribly ripe issue in a sense of having been able to work through all the allies yet and figure out I'm sure there's a lot of divisiveness in terms of how NATO should get back involved in Iraq given how divisive Iraq was so I do I think he wants to stay focused on that it's a Russian message and it's not quite ripe yet for a summit Great, right over here Thank you Stefan Gorbel, EuroNews following up on this, would you exclude in the long run any combined NATO action against ISIS? I mean there are some countries that have interest in the United States, United Kingdom obviously Turkey of course even Germany that promised to sell weapons to the Kurds is that totally excluded or thanks? I don't think it's excluded, no I again it's not on the summit agenda but it's a light motif if you will that will be happening we're living it in real time the conversations are happening bilaterally and perhaps multilaterally in various forms certainly through the UN form as well so NATO is going to be a piece of that and I think that's something for the NAAC the North Atlantic Council I'm sure is thinking through how do they put this on their agenda how do they address this it's been on their agenda in terms of the Syria piece of it but now that it's more of this Islamic State transnational crisis I have no doubt that NATO will take it on whether that means there would be NATO sanctioned action a la Libya I think is open-ended but I wouldn't take it off the table Great, in the back Thank you, Leandra Bernstein-Rio Novosti I wanted to follow up has NATO NATO allies anticipated the Russian response to a permanent rotating presence which is east of the former Iron Curtain and then what's the strategy to deal with that or will they continue to just respond to crises after crises is there a strategy and then the second question is your thoughts on the decision by Putin not to attend the conference Well, the allies have already fully anticipated a very strong Russian reaction I'm sure following this in fact this is what's caused some hesitancy by NATO allies to actually put forward a more robust presence because they are very fearful of provoking Russia I mean again, as an analyst I return to the reason the only reason that NATO is doing this is in response to Russia's actions this is not a unilateral NATO decision came out of the blue it is in response to instability on NATO's border and that is seen as not as a provocative act seen as reassuring allies and sending a clear message that NATO will defend its members but I am sure the reaction will be strong it will probably again provoke the additional Russian forces along in this area so we will see a build up on both sides absolutely and this is what many analysts and the media will say back to the Cold War there are Cold War elements of it and this is going to be one of them where you will see a response but we have to react to the instability and quite frankly it ensures stability it does not increase instability again as I was reviewing and looking at even just in 2006 as I said the last time a US president was reading some of the press clippings of the briefing the White House had done before that trip in 2006 it was all about how to form that NATO-Russia partnership so it is amazing to me we were talking about how to create a stronger NATO-Russia relationship and literally over the last six, nine months the picture is now just so completely changed NATO has tried I am sure with many faults and problems that not a perfectly implemented policy but NATO has tried to be a partner with Russia and clearly that has failed I just want to add one thing on that I think the fact that the NATO approach will be focused on NATO countries so this will be about reassurance inside of NATO's borders and very explicitly taking a step down to a building partnership capacity approach to the non-NATO members is a way of signaling to Russia exactly that defensive intent if you will the types of equipment the types of exercises, activity that NATO rolls out as part of this approach will matter in terms of the signaling to the Russians they can choose to message it the Russians can however they like but I am sure NATO will be working hard to determine what looks defensive in terms of a rotation and what could be misinterpreted and thus lead to a miscalculation that isn't helpful to anybody Thanks Secretary Gates at his last speech to NATO used some very harsh language about this talked about a two tiered alliance and all of that I think recently the vice president has said things about this Secretary Hegel has reiterated this and as you mentioned only four countries are living up to that commitment in light of our own drawdown of forces we are tired after 10 years of trying to push this message of trying to push these NATO countries to step up their commitments how hard are they going to push and are any of these countries you mentioned Poland does anybody else likely do sort of try and step up to the plate So to tag team on this in many ways if your talking points aren't working is a time to change your talking points and this has been a 15 year you are absolutely right Secretary Gates farewell message you have to do something it's a crisis that Europe in some ways is demilitarizing I think it has been Ukraine crisis has been a wake up call now whether the Europeans will hit the snooze button or not again I don't know but it has certainly shaken them that they have allowed their military defense spending to atrophy to a point where they are now vulnerable and they did understand that US the US presence in Europe was rapidly diminishing now again things have changed dramatically we are going to have a presence in Europe but they have to step forward I agree with Kat I think this whole 2% you can spend 2% but as the Greeks do it's for territorial defense it's not an asset that the rest of NATO can use but again Estonia is not going to compensate for the lack of 26 other allies from stepping forward so it has to be meaningful it has to be purchasing NATO interoperable equipment that can be used both for collective defense purposes but also for crisis management and this is where the ISIS role or what have you NATO can't swing totally from it's been out of area for 13 years we are going to go only for territorial collective defense NATO has to keep that full spectrum fewer and fewer allies are able to have that spectrum so in this way for many many summits these defense capability initiatives smart defense this and we keep rebranding it but there is very little to point to so I really hope now is the time that they realize no more we have to put forward and we have to get serious about this it has started a conversation that I have not heard in a long time in Europe but let's all be clear Europe's economy is not only fragile I think it's highly vulnerable and there is going to be limits to what Europe can do I agree with Heather I think the question is not whether burden sharing however nicely put or smart defense which miraculously disappeared off of the lexicon in the last several months are put forward there will be a message about sharing burden sharing costs it's whether the 2% piece continues to be the particular 2x4 used I think the United States will continue and others who are spending will continue to be pressing allies to commit on their defense commitments but whether that constitutes a 2% or a different approach I think is the issue what I find promising or encouraging about this readiness action plan is the pressure that is the press inside the defense community across NATO because there is a lot of agreement at the defense level across NATO it's really the defense community to the political community where there is a lot of variation across the allies the press there will be on concrete initiatives that show capability and that's where something like the readiness action plan could prove out if it's just the US and the Germans and the French that's not a great step forward for the alliance but at least it's somewhat reassuring to the Eastern European allies but if through this action plan and other concrete steps they start to pull in capabilities from some of these other countries that have been less likely to be contributors that will be meaningful in terms of a summit outcome in the back here thank you Inga Czarny for the Polish Press Agency in the light of Ukrainian crisis and this permanent NATO basis in Central Europe unnecessary of this spearhead force as Secretary Rasmussen yesterday announced would be efficient you know I think what NATO is trying to address is the fact that Russia has demonstrated very ably rapid military mobilization and that is the area where NATO has is not as where it needs to be in its own within its own alliance so the idea of prepositioning or putting more on a rotational basis ready forces is in some ways to respond to that mobilization again it's a defensive it's not an aggressive it's to be ready and the focus of the incision in Poland the multinational core northeast will be to be both land obviously air and sea so it's to look and engage at how NATO can be ready and so in addition to the session you'll also see a much greater focus on the NRF NATO response rapid forces and it's all about quickness and having equipment preposition there to be able to use again we have not believed this has been an issue for a decade plus and NATO is playing catch up a little bit but it's not a provocative step I just to highlight because Estonia is literally almost a stone's throw away from Finland just this week alone there have been two Russian air incursions into Finnish airspace you had yesterday the Finnish Prime Minister and President starting to openly talk about NATO membership again never contemplated but we have to look at this from a regional perspective there are vulnerabilities here and this helps both NATO members but as well non-NATO members in understanding that there's a presence there to support their very shifting security environment let me just add the types of threats that NATO feels it needs to improve itself upon are these hybrid threats these asymmetric threats these unusual unconventional approaches whether it's from Russia or from elsewhere in the world but the Russians have certainly in the Ukraine crisis shown themselves quite adept at those and certainly Estonia itself felt the cyber end of that ago so the types of capabilities that NATO needs today are going to just look different obviously than they would have during the Cold War so you wouldn't be looking for large formations of ground forces you know rolling in and stationing themselves in Eastern Europe that's not the most effective use of NATO capability today and just as Heather said it's about that rapid reaction capability the NATO Response Force the US committed in 2012 a stronger commitment to the NATO Response Force there's a sub-element in that called the Immediate Response Force the IRF which I think is what NATO is really going to be focusing on here which is a very quick reaction element in the NRF along with the aircraft the tactical aircraft that are stationed elsewhere in Europe can they come in and be positioned in crisis and then of course the naval access on the Baltic side where we already have some US NATO capabilities there so it won't look like sufficiency today looks a lot different than maybe one would have thought of in the Cold War we are not looking to go to war with Russia we are looking to prevent any kind of unusual unconventional approaches that might come after NATO territory or NATO stability and look for ways that NATO could help those alliance members feel secure and actually be secure I'm Laura Sarekowski from the Finnish newspaper Helsingin Sanomat I know that the US of course respects every country's decision of making their own decisions as far as joining NATO or not but from the US point of view does the neutrality of Sweden and Finland has it lost its credibility well let me begin by saying both Sweden and Finland have been extraordinary partners to NATO joining in Afghanistan Sweden and Libya Operation Unified Protector so we have had extraordinary partnership between NATO and Sweden and Finland clearly the situation in northern Europe has begun an incredibly active debate within Finland I would say the conversation is much more robust in Finland today than it is in Sweden perhaps because of Swedish elections on September 14th and that needing to resolve itself before Swedish government really thinks through these issues but it is for every country to decide and this is why NATO the open door policy it is open but I will say that some have commented it's awfully difficult to get home insurance when the neighborhoods on fire you sort of wow is this the right time to start focusing it on now I think there are some in Finland that argue this probably should have been thought through a little bit prior to the current crisis but I think there is an active dialogue between Finnish defense officials and NATO officials also bilaterally between the US and Finland there is a stark difference between a NATO member and a non-NATO member on defense full stop but we do highly value our partnership with Sweden and Finland and again as you look at this regionally there of course is a vulnerability of non-NATO members ironically 10 years ago when the Baltic states were joining NATO Finland and Sweden thought their strategic vulnerability was the Baltic states and I would tell you today I believe their strategic vulnerability maybe Sweden, Finland's non-NATO membership so the world turns so I think we'll follow obviously the Finnish debate quite closely and stay in close touch on a mill to mill basis great Jill thank you Jill Doherty soon to be at the Wilson Center I just wanted to ask following up on the Finnish question I think is really interesting you said there's no appetite for expansion of NATO but we do have this action by Finland and I just wanted to ask I'm presuming when you say there's no appetite it means Georgia Ukraine now Finland I mean if Finland said we want to join wouldn't that be a game changer how would NATO really realistically react or could I delve into what you were just saying when the neighborhood's on fire etc. what does that mean exactly I mean they wouldn't say no would they Jill that is a great area to explore and actually CSS has been doing a lot of thinking about how that would look you're right I think there is a stark difference among the NATO allies as they're considering Montenegro's membership actually Bosnia Ukraine and Georgia and then thinking about Sweden and Finland they're just two very they're at very different places in their approach I think you'd see where Sweden and Finland again working so closely with NATO have in some ways their military modernization and their close relationship puts them ahead of you know the mil-to-mil dialogue with other NATO aspirant countries for sure would it be a game changer if Finland and Sweden were to formally come to NATO and seek membership yes it would give the alliance an extraordinary conversation about what that would mean speaking of provoking that would certainly provoke a very strong reaction from Russia and then there are actually and I'm not a specialist in how this would work but it would actually be the process of it would take quite a while before all NATO members even if a formal invitation was provided by NATO it takes all member states have to ratify the U.S. Senate would have to ratify an amended NATO treaty Belgium has 16 parliaments alone that takes a while so would article 5 begin at the moment of invitation would article 5 only would be a series of this but that would be part of the conversation and can you make that decision at a moment of great crisis you know this is an analyst dream we could spend an enormous amount of time trying to spin the scenarios out but I think your question Jill is this is the dramatic change that I'm talking about this is the dramatic thinking we need to do and this is the U.S. policy needs to be much more broadly thinking about this from a regional perspective but it's an extraordinary challenge for sure great question we have a board game on this upstairs if anybody wants to come and play one day we're there we go David Ivanovich with Argus media despite the difference of opinion regarding sanctions against Russia do you expect the U.S. and the EU to use this summit as an opportunity to impose new sanctions on Russia I don't believe there will be a concerted effort as there was a few weeks ago to do a new round of sanctions clearly I think there will be some focus on next steps as I look broadly over the last several weeks the more we've imposed sanctions the greater the crisis has escalated quite frankly so we're seeing an inverse reaction we wanted to change Russian behavior and action and in fact if anything it's escalating so what happens now I'm sure every capitals are trying to digest what happened yesterday and these long talks between presidents Poroshenko and Putin you know what this peace plan or the ceasefire will look like but I my own sense because of the escalation because of yesterday's news of this new opening of separatists in the Mariupol area that to me is actually a pretty significant event that if there's an access a loss of access to the sea that we're in a different place as well so I'm sure there'll be monitoring events I think there will be preparations to see what more can be done if the crisis continues to escalate but I don't sense you'll get anything more on another round of enhanced sanctions I don't sense that's being prepared right now great straight in the back right here you mentioned that thank you so that you mentioned that you mentioned that open door policy is still on the table but enlargement is not but you'll never about Finland and everything but about other countries from Balkans maybe more about Montenegro Macedonia is really tough position for years Bosnia has to go probably is not ready but can you elaborate a little bit about Montenegro because Montenegro is almost ready but they have big impact from Russia especially their secret agencies and everything can talk about that please thank you and I'm sorry I failed to name the Republic of Macedonia is also an aspirant country that had been invited in 2008 depending on resolution of the name issue there was a great frustration when NATO signaled several weeks ago that this was not going to be a moment where an invitation would be provided to Montenegro you're absolutely right there has been an extensive amount of work and preparation for that eventuality you know as Kath was saying the preparation for a country a NATO aspirant country to join NATO happens at two levels obviously the military to military the preparation of that ally to be able to contribute to NATO but then there's also that political track which means that that ally that potential ally is ready to accept the responsibilities of joining value based alliance and then of course there is the political willingness by all members to accept that member and that is where the political willingness and the political will of NATO over the last several years has diminished greatly in some ways that's because the table has grown so large NATO at 28 is a very challenging alliance to get consensus on 28 where the threat assessment exception amongst those 28 are very different and trying to get that alliance to contribute and to every member to reach that 2% goal or to make that commitment so I think people were hoping that the United States would be much more forceful and it's pushing the enlargement agenda that has been the historic experience I don't think you saw a dramatic push from Washington and there's great reluctance in Europe to do this but what that means particularly for the western Balkans that is unfinished business in Europe and we know when we neglect unfinished parts of Europe we have crisis and in some ways that is Ukraine that is Georgia so the alliance will need to refocus on this because now what you have are frustrated aspirant countries that have been working hard on reforms that may start backsliding and I feel that that door is truly open it's only open in theory so there has to be much more important thinking done about western Balkan enlargement for both NATO I would also argue for the European Union as well to make sure the western Balkans has not become a crisis I'm sorry about that do you have any information about that? I don't have a great sense of that I mean watching just on the EU side Serbia as an EU aspirant country has said they will not impose sanctions on Russia this is going to be a very difficult choice for some Baltic countries that rely very heavily on Russian energy Russian financial support and so the tug and pull that you're seeing in Ukraine between Russia and the west will continue to play out in the western Balkans and as I said unfinished business when it's too difficult for us to tackle allowing it to just be put off the agenda will return to it at great price I have time for about one more after that Karen Blair Brand with the Baltic American Freedom League what's the perspective on the US and NATO with regards to the French sale of warships naval warships to Russia Cameron is against it and I'd like to also know this is set sale for the Baltic and Black Sea Seas actually in November and I also hear from the Baltic American Freedom League that soldiers are being already trained from Russia on these French ships in France Yes the status of the French Mistral sale to Russia certainly from President Obama every senior official has certainly registered its deep, deep concern about this sale unfortunately it has not had has not led to a change of decision by the French government right now I believe approximately 400 Russian sailors are being trained in France now and the first ship I think named the Vladivostok will actually go to the Pacific Fleet the second vessel though to be delivered ironically is going to be named the Sevastopol and will be delivered to the Black Sea Fleet this is something that we're going to have to continue to work very hard to convince Paris that this is absolutely the wrong decision unfortunately they're clinging to the economics of this and the fact that they would have to pay Russia to deliver the ships I understand about $1.2 billion so and if you've been following the French economy you understand that that's a very difficult decision we're going to have to keep working on it a lot of a pressure is going to be has to be applied unfortunately the first vessel is due to be delivered in October so we're running out of time it certainly impacts NATO solidarity What about NATO possibly purchasing them there have been some creative thinking about that but I don't think that's been I guess I said what about NATO possibly purchasing them helping solidify solidarity I'm sure there's been some creative thinking about how to help our French friends get to the right place on this decision I don't think there's been anything that has come forward on that as of yet I just want to add that the global arms market has changed dramatically for the last ten years the Russians are the second largest exporter next to the United States of arms it's going to be very tricky let alone their energy sway if there continues to be tension between Russia and the rest of the west so this is one example the US has its own which has been the MI-17 helicopter it needs for Afghanistan for the Afghan forces which is Russian made these kinds of things are going to pop up all over and there will need to be an alliance approach to how it deals with Russian sold arms great I want to thank everybody for coming this morning we'll have a transcript of this briefing out hopefully by close of business today it'll be at csis.org you can follow us on twitter at csis and please let us know if you have any questions for our experts we want to thank Heather Conley and Kathleen Hicks for this wonderful briefing thank you


So, formally welcome everyone to your exam preparations. My name is Elizabeth Boyd, those who haven't met me before. Before we start with the session, please make sure that you complete the register, which I will post in the chat. Go find that register, which we will post in the chat. I have shared it in the chat. Please make sure that you complete it. Every week you come in, make sure that you complete that register. If you can see the register, please let me know so that I can I can repost it again. And that it's very important that you complete the register. And yeah, today's session, we're going to do an exam preparation by looking at different questions on how to solve fractions, how to solve things using lowest common multiples. And how to solve problems with powers and so on. How do you guys do the basic numeracy introduction questions that you get in your exam paper, which from part mostly of question one up until question five or 10, something like that. If you need to send me an email where you want to ask some other questions or you are unsure of in terms of the your numeracy, you can send an email to eboy. And if you have any technical issues, where to find the notes or there are no notes, so you won't have notes, but where to find the recordings. The same link way they shared with you probably they should be the schedule where you are able to go and find the recording. So the recording will be uploaded onto the website. So if you can find them, send an email to CT and CT and touch at unisa.ac.za and they should be able to send you the link on where you can access the recording for the session. Yeah. So that is it. So I'm going to share again with you. I'm going to stop sharing. I'm going to share with you all the, for some reason now, since I am the administrator, I can share the file through anyone of you who have already downloaded those who joined earlier today, if you are able to share, please share the files with everyone again because I'm unable to share or attach documents. It doesn't allow me to share a document. Is anyone able to share? Allow me. Unfortunately, because I am unable to share the same documents again, those who joined late. I don't know how you're going to receive the files, but anyway. Let's go and use the 2019, I think, 2019. 2019 6 E1. We will use that one. If we're done with it, especially the topics that we need to be doing today, once we're done with that, then we can go to the other three, the other two, which is 2018 10 and 2018 6. But we will start with one of those exam papers. So let me share my screen. And because this is exam preparations. So I'm not going to be doing some lot of explanations. You are going to be doing the lot of calculations and lots of activities. I will just be here to support you as well. So let's kick off. I'm not going to explain the type of questions that you're going to have and all that because I don't have access to your current exam guidelines. And you should receive that from your lecturer. My task today is just to look at equations and topics. So since we're looking at topics with fractions, topics with lowest common multiples and topics with powers, we can also we're going to skip the expression questions for now. We will do the expressions and equation later on next week in the next week session. So we're going to skip all this. We'll go to the fractions. So if questions like this comes up, you need to be able to know how to work with fractions. Right. And especially you need to remember and always know your botmas or your basic operations, how they work and the botmas rule as well. Because botmas rule says brackets first, operations or powers, division and multiplication, addition and subtraction where division and multiplication have the same priorities. You can work from left to right together with addition and subtraction have the same priority work from left to right. You also need to consider that when you work with fractions. Number two, when you work with fractions as well, you need to consider the following that in terms of addition and subtraction of fractions, if they have the same common denominator, you just add or subtract the numerator. What do I mean by numerator and denominator? So with fraction, the top parts of the fraction, we call that numerator. The bottom part, we call that denominator. You also get different types of fractions. I will get to that just now. So if you have addition and subtraction of fractions, then you need to check if they have the same common denominator and if they have the same common denominator, you add or subtract the powers. If they do not have the same common denominator, you're going to find the common denominator and there are different ways of finding the common denominator. And then once you have that, you solve your fractions. If you have multiplication and division. Now, multiplication, let's start with that one. For multiplication, there are different ways of multiplying. With multiplication, if you have fraction with A and B, multiply by, let's call it AB and AB. So, maybe let's call it C. AB, let's call it C. A divided by B and C divided by A. Let's say this is our fraction. Now, with multiplication, there are three ways that you can do your, you can solve your fraction. You can multiply the numerator with numerator, the denominator with denominator. So you will say A times C, that will give you AC. B times A will give you AB. And that's how you can solve your equation by doing numerator, multiply by numerator, or denominator, multiply by denominator. That's one way. The other way, you can simplify the equation by please crossing. If the numerator and the denominator can simplify, meaning they can go into each other, you can simplify that. If A can go into A, or A at the bottom can go into A at the top of the opposite.  And then you can solve the equation. We're saying one times C, which will give you C and B times A. Oh, sorry, B times one will give you B. Remember as well, the first one we said it is AC over AB. You need to make sure that when you solve fractions, I must just go back to this one. You need to make sure that when you solve fractions, you simplify them to their lowest form. Therefore, it means you need to make sure that you cannot do anything to that fraction, anything like that. So with AC divided by AB, A and A will cancel out and you will be left with C over B. And that is the lowest form. The same way as here, you will have your C and B, they cannot divide into each other. So if A and A cancel out, you are left with C divided by B and hence the answer will be C divided by B. That is one way of doing your fraction. The last way of simplifying fractions is that, so we have A divided by B times C divided by A. The other way of simplifying fraction is to see if you can simplify up and down. So it means if A and B can be simplified, you will simplify that. If C and A can be simplified, you will simplify that further. So those are the three ways of answering multiplication. So which is the thing that you need to always remember. The last thing is your divisions. So in terms of division, so let's say we have A over B divided by A over C. So now when you have an equation that has a division, we apply what we call it KCF. Keep the first equation, change the sign and flip the second equation. What do we change the sign to? We change the sign to a multiplication. We flip, the numerator becomes denominator, denominator becomes numerator. So we keep A over B. We change the sign to a multiplication. We flip C will be divided by A. That is when you have fractions, the way you need to make sure that you are finding the quotient or the division of the fractions. Simplifying them means you're going to apply the KCF method. And once you have applied the KCF method, then you can do your multiplication. But always remember, both must rule applies. Division and multiplication have the same priorities. If you have an equation where it has division and multiplication together, you just apply it from left to right. So let's say, for example, this is the equation that we are looking at. You can see that the equation has a division and a multiplication. Remember now, there are three fractions. The KCF says keep the first where there is a division. You keep the first, you change the sign and you flip the next fraction, not the other fractions, right, after that. So it means you're going to use your KCF on this and multiply by 3 over 7. You keep the rest. If the division was on this side, then you keep the other things the way they are. So if we say there was a fraction like that, it's multiplication and division. You just apply the keep, the change the sign and the flip on the one. So I hope that makes sense. So let's go through and work through this exercise. And then we will find more exercises that you need to do yourself. So simplify the following expression as far as possible. As far as possible means until the fractions can now or no longer or cannot be simplified. Therefore, it means if you can still divide by a second number, the fractions, you simplify that. So let's go on and do this one activity. So we have division and multiplication. What must says division and multiplication have the same priority. I need to work from left to right. So because it's division and multiplication and a division eventually changes into a multiplication. Then I can just do the KCF. So KCF will say 7 divide by 24. I must keep that change the sign to a multiplication and flip 14 divide by 12 multiply by 3 over 7. I've applied my KCF. Keeping the first change in the sign flipping the next fraction. Now I need to simplify because it's multiplication. And now you need to take into consideration that this is all denominators and numerators at the top. And things can, the first fraction can also simplify the last fraction or the second fraction or vice versa. So yeah, we go into apply the method of simplifying first because if I need, if I say 7 times 14 times 3, I'm going to get a huge number. And then I'm going to get 24 times 12 times 7. I'm also going to get a huge number. And then I need to think and figure out can they still be divided? Can they still be done this? Can this build the, you understand? So in order for us to simplify the equation for that, we can say 7 and 7 can cancel out because 7 goes one time into 7 and it goes one time into that 7. The other thing, we can divide into 12. 3 goes one time into 3 and it goes how many times into 12? It goes 3, 6, 9, 12. It goes 4 times. So it goes 4 times. So I keep the 4. And 14 and 24, something can divide into those. What is the number that can divide into 14 and also can divide into 24? I think the only number at this point is 2. How many times 2 go into 14? 2 goes 7 times into 14 and it goes 12 times into 24. Now I have 1, so I can write out what is left. 1 over 12 times 7 over 4 times 1 over 1. That is what I have right now. In order for me to simplify this, then I can do multiply everything that is at the top with everything that is at the top, everything that is at the bottom with everything that is at the bottom. So 1 times 7 times 1 is 7. And 12 times 4 is 48. Can 7 divide into 48? No, it cannot. It leaves a remainder and therefore the answer is option 2. As I said, simplify your answer as far as possible. And that's how you will solve infraction. Are there any questions? Before I give you an exercise as well. If there are no questions, you see there is a... Thank you, Carol, for sharing the document. Okay, so let's move on to the next fraction. Simplify the following fraction as far as possible. Now, looking at this fraction, it has a subtraction and a plus. But must says addition and subtraction have the same priority. Because but must addition and subtraction have the same priority. Therefore it means we went from left to right. And addition and subtraction of fractions, we first find the common denominator if they do not have the same common denominator. The denominator is the value that is at the bottom. If we look at this, 6, 4 and 24 are the values at the bottom. They are not the same common denominator. And what is a common denominator? Common denominator is a number that it can divide into the other numbers and not leave a remainder. So if we're looking at 6, 4 and 12, what is that number that can divide into three of them and not leave a remainder? What is that number? Let me ask you that question. The question, the common denominator? No, all four of them needs to divide into that number. Sorry, not what is the number that can divide into all this. All, sorry, it's me. I explained it all wrong. What is that number that all three of this denominator can divide into and not leave a remainder? That is what I should have said it. A common denominator is that number that all the other numbers can divide into and not leave a remainder. So we have 6, 4 and 24. What will be the number that 6, 4 and 24 can divide into and not leave a remainder? 24. It's 24. So our common denominator will be 24. So you're going to write your common denominator 24 at the bottom. And once you have your common denominator, all what you do is simplify. Simplifying will say 6 goes how many times into 24? It goes 4 times and you multiply. So you will say it goes 4 times and you multiply with a 5 minus. So you're going to multiply with the numerator and you go and say 4 goes how many times into 24? It goes 6 times and you multiply with 3 plus. So you will keep the sign. 24 goes how many times into 24? It goes one time and you multiply with 7. And you can simplify. 4 times 5 is 20 minus 6 times 3 is 18. 1 times 7 is 7 divided by 24. Remember what my rule says? Addition and subtraction have the same priorities so I can work from left to right. So we keep from left to right 20 minus 18 plus 7 equals 9. The answer today will be 9 over 24. Is there a number that can divide into 9 and also divide into 24? 3 can go into 9 and it can also go into 24, right? 3 goes 3 times into 9 and it goes 8 times into 24. And the answer will be option 3. You see how easy it is to solve fractions? Very easy. Are there any questions? So let's move on to the next other fraction. I'm going to have to let you do this one on your own and then we can check if our understanding are correct. Remember always those who attended my sessions, they know that we always use Newman's problem solving framework or method, where it says ask yourself. What is the question asking you to do? It's asking you to simplify this as far as possible. What are the things that they are giving to you? What are the facts given in this? You have addition, multiplication and subtraction. What are the things that I can use or apply to help me solve this? But my rule, multiplication, addition and subtraction. So multiplication comes before addition and subtraction. So therefore it means I'm going to solve first the fractions where they are multiplying before I can do the addition. Okay. And then what now? Then I will be left with addition and subtraction. The Bodman's rule says they've got both have the same priorities. Therefore I'm going to work from left to right. And then because there are addition and subtraction, what is the data I need to do? I need to find the common denominator and then you go find the common denominator between the three fractions and then simplify. Now I'm leaving it to you. And you're going to keep three over four plus and minus one over three as they are and only solve what is in the middle. That is your first step. So your first step is going to solve this and get the answer and then solve the next. We remember solving multiplication. You multiply what is at the top with what is at the top at the bottom with what is at the bottom. And if you are able to simplify, you can simplify top top or diagonal diagonal. I'm giving you two minutes while I go get myself what I will be back. How are we doing? Are we winning? Is it better? Is it getting there? Ready? Yes, I've got it. What is your answer? One and one over 24. One over 24. So someone knows their way. Okay, so let's help those who don't know how to get there so that everybody is on the same page. So what is the first step we do? We keep multiplication. Yes. So I'm going to keep three over four. Plus, how did you solve that? It's five over eight. One times five is five and two times four is eight minus one over three. What is the next step? Lowest common multiple. What is, yes. 24. That will be 24. Four goes how many times into 24? Six times. Six times three? Eighteen. Eighteen. Eight goes how many times into 24? Three times. Three times five? Four of ten. Three goes how many times into 24? Eight. Eight times one? Eight. Eight. Eighteen plus fifteen minus eight. Twenty-five. Twenty-five over twenty-four. Now I must explain the types of fractions that we have. In fraction we have what we call a proper fraction and an improper fraction and a mixed fraction. So now a proper fraction it is when your numerator is small and your denominator is big. An improper fraction it is when your numerator is big and your denominator is small. And an improper fraction is not your lowest form of a fraction. There is another way of representing fractions so that you are able to use them in your calculation or when you're simplifying them. You can use the improper fractions. A lowest form of an improper fraction is what we call a mixed fraction and a mixed fraction has a whole number and a small and your big number at the bottom. So to get to a mixed fraction you need to simplify this which will give you a whole number and a remainder. And then you keep your denominator as you have it and your denominator will become a small or sorry it will become a big value once you have simplified that. So how do we simplify? We say 25 goes how many times into 24. It goes one time and the remainder is one over and then we keep the numerator as it is and hence it is option four. Let's carry on with the next one. Now we've worked with numbers now if we're going to work with we're going to work fractions with variables. Remember variables are placeholders but we need to be able to also know how to simplify the placeholders if they are in a fraction. And that's where what we call lowest common multiples came in all lowest what are the common multiples and all that comes in. So in order for us to simplify variables, we need to find the common multiples or is the same as the multiples of those variables. So if we have variables like B and B squared is the same as B multiplied by B, right? If we have a variable like AB is the same as A multiplied by B which are called multiples. So in order for us to find the common denominator or the common multiples, we need to find the multiples of the values that we have. So at the moment we have, I just need to do this so that I'm able to write the values correctly. At the bottom we have A, we have B, we have B squared. Oh, sorry. That's a 4B. You also need to take that into consideration. The 4B. In order for us to find the multiples, the multiple of A is A. Multiple of 4B is 2 times 2 times B will give us the multiples of 4B. The multiple of B squared is B times B. Now to find the lowest common multiple, we're going to take the values that we see here and only keep the value once. If, sorry, if the value already exists in the multiple, we don't include it again. But if there are two of them, then we have to keep the other one as well. So let's look at this. So we have A multiplied by 2 multiplied by 2 multiplied by B. Coming to the last one already we have a B, so we cannot include the B again, but we also have 1B. So this B is not considered and we're going to multiply with the B. And hence we'll end up having 2 times 2. We always write the number first. It's 4AB squared. That will be our lowest common multiple of AB and B squared. That is if we solve the equation like that. I was just explaining and demonstrating to you. However, we need to take into consideration as well what we have here. We need to take that into consideration. It's division, multiplication. Because it's division, division we do KCF. So we need to first change this before we go find the multiples. So let's first solve this. We keep the first, which will be 2B divided by A. We change the sign, which will be multiplication and we flip. It will be 4B divided by A and A cubed divided by B squared. Now we have our multiples here. So we have A, A and B squared. Find the multiples A, AB times B. Therefore, if we need to find our LCM, we say A. We already counted A. So therefore A, the second A won't be considered times B times B. So because the two B's are on the same line, we need to consider both of them. If there was a B before, we would not consider the first B. So because there was no other B, so we keep everything as is. So our lowest common multiple here will be A times B squared. That will be our LCM, AB squared. Now, how do we solve these fractions like this? So going to say A goes that many times into AB squared. Therefore, it means A will cancel with A. You will be left with B squared times 2B. Come on. Times 2B. Oh, sorry. What am I doing wrong with all this? You can't find the LCM for a division and a multiplication, guys. That's one thing that I'm teaching you which is wrong now. You find the LCM only for addition and subtraction. We need to always remember that. Multiplication and division we simplify. Now, sorry, that's my bet for not reminding you of that. When we work with multiplication and addition, we need to remember the powers, the rules of the powers. It says if I have A multiplied by A, A is the same as A to the power of 1 and A to the power of 1. And if I have a multiplication of the same bases, they've got the same base, then I add the power. So it will be A1 plus 1 which will be equals to A squared. If I have A divided by A, A and A have to the power of 1 and it's a division, they've got the same base. Then with the rule says we're going to subtract what is at the power A1 minus 1 which is A to the power of 0. Any number to the power of 0 is the same as 1. So I've already touched the powers. I've already touched most of the things that you need to know about the powers. What else is there? The other thing that is there is if you have A to the power of 2, 2 to the power of 2 is the same as A to the power of 2 times 2, you've just multiplied the powers which will be A to the power of 4. If you have a fraction, A to the power of a half is the same as the square root of A. Those are the things that you need to always and constantly remember when you solve fractions. If they give you a root, you need to know if they give you a root of A.  Sometimes they don't write it for a square root but you must always know that there is that invisible 2. Same way as there is always an invisible 1 for any number where there is no power. So there is an invisible 2 there and the square root is the same as 1 over and 1 over the 2 in front of the power. So what if they wrote it like this? If they give you the square root of A, sorry, the cube root of A. So because it's a cube root, we know that the root is the same as 1 over, so the denominator will just be 3. The same way if they give you the quadruple root of A, it will be A to the power of 1 over 4 and so on and so forth. So those are the things that you always need to take into consideration when you solve multiplication or division of powers. Say different, difference true when you have addition and subtraction. So those are the basic concepts that you need to learn. Now let's go to this. We have division, multiplication, we do KCF on the first one, KCF. We keep 2B over A, we change the sign to a multiplication, we flip 4B over A, we multiply by A cubed over B squared. What's next? Next, we have a multiplication. We've learned about multiplication of equations. Remember, multiplication, you can multiply what is at the top with what is at the bottom with what is at the bottom. That's what we did. We also said something to the effect that you can cross or simplify. So for example, if you look at A cubed and we spoke about this, it's A multiplied by A multiplied by A. So I can rewrite A cubed as, I can rewrite it as A multiplied by A multiplied by A so that I can simplify it with those other ones at the bottom. And B squared, I can also simplify it and write it as B multiplied by B in terms of multiples, right?  So yeah, we have 2B and 4B. So we can also say B and B will cancel out or let's leave it like that. I just want to demonstrate something as well that how you can also simplify this. Now, we can also multiply what is at the top with what is at the top and continue. So we already canceled out the A's. We're not going to get them back. The reason why I'm leaving the 2B and the 4B is so that I can explain something when we get to the end. So we're going to keep 4 times B like the way they are. So 2B times 4B is 8B. Remember, they've got the same base, B to the power of 1 and B to the power of 1, right? So 2 times 2, we multiply numbers as numbers. It's 8. B to the power of 1 plus 1 will be B squared, right? I'm just going to keep B squared the way we have it so that I don't have to do anything to this B at the bottom. And A is what is left at the top. Divide by... We didn't have to even simplify it to B times B so we can put it back to B squared because it doesn't really make any difference. And now, B squared and B squared will cancel out. You will be left with 8A. That is one way. The other way you could have done it is to say B cancels out with B. And B cancels out with B. Then you are left with 2 times 4 is 8A. And at the bottom we are left with 1, which is the same as 8A. So I just wanted to show you and demonstrate to you that you can do it in many different ways and still get the same answer that you have. So I know that I'm complicating things, but that's the purpose of skills literacy. It's to show you many different ways of untieing things so that you feel comfortable to choose whichever one you want to use. Are there any questions? Before we go, find more fractions. Are there any questions? No questions and nobody is saying anything to me. We're almost left with 35 minutes. Let's look at no questions. Okay, so now we have subtractions. And remember with subtraction we said we find the LCM first. Right? So you can apply the LCM. So let's go and apply the LCM. We have 5AB and 2B. What are the multiples? 5A times B, 2 times B, LCM will be 5 times A times B times 2 and we already have B so we don't have to read another B. The answer here 5 times 2 is 10A times B is AB. So our lowest common multiple will be 10AB. Now we're going to say 5 goes how many times into 10 because a number can divide into a number, not a variable. So 5 goes how many times into 10? It goes 2 times. 2 times 3 is 6. A goes how many times into A? It goes one time so there's nothing. B will also cancel out with B so the answer there will be 6 minus. Then we come to the top one, to the next one. So the next one is minus 2 goes how many times into 10? It goes 5 times. There is no A there therefore we're going to keep A times 1. We're going to keep that. 5 times 1 is 5 and because there is no A so we're going to keep that A there. B goes one time into B so then it's nothing. So the answer is 6 minus 5A divided by 10AB. That's the answer. Which will be option number 4 and that's how you will simplify and answer the questions. Are there any questions guys? You are so quiet. Okay. So that covers almost everything that I wanted to showcase and show you how to simplify fractions. With powers, lowest common multiples and so on. So we have covered almost 1, 2, 3, 4, 5, 6. 6 questions that will appear in your question paper probably. The others are more about expressions and equations. We will solve them next week when we deal with that. Okay, so in the next session or next question paper we can look at 2018-10. So let's look at similar questions and on this one I'm going to start first with the fractions with letters. Because I think we did a lot in terms of those with all these fractions. Okay, we can start with this one. Even though it's not visible but at least it's something. So this is 18x squared divided by 4y minus into bracket. 3x divided by 4y multiplied by 9. Now, but must always important, very, very important. But must brackets first. Everything with the bracket must be solved first. So in the bracket we have 3x times 3x over 4y times 8 over 9. Now remember multiplication you can simplify numbers can simplify numbers. So here we have 3, 3 goes one time into 3 and it goes 3 times into 9. If we simplify this it will go 3 times into 9. 4 can go one time into 4 and it goes 2 times into 8. And then we can simplify what is inside the bracket which is 18x squared divided by 4y minus. What is inside the bracket 1 times 2 is 2 over 1 times 3 is 3. So now you have two fractions. Remember always this must always be your go to LCM because we've got fractions. Sorry, letters, variables. So 4y and 3 is the same as 4 times y. So we can say our LCM will be 4 times 3 times y which is 12 times y. So your lowest common multiple will be 12y. Simplifying 4 goes as many times into 12. That will give you the answer. It goes 3 times. 3 times 18 is 54, right? That will be 54 and y goes y times. It goes one time into y. So then we are left with x squared minus 3 goes as many times into 12. It goes 4 times 4 times 2 is 8. What is the remainder here is y. So you also need to keep that into consideration when you answer this question. There might be some error with this equation. It might be that this is a division and if this is a division, then it changes everything because I can see that I'm not seeing the answer here. I'm going to convert that, especially with the past exam papers at UNISA. You must be very careful with them. The dot sometimes disappears when they scan the exam papers. So we're going to treat this as a division and multiplication. And therefore if it's a division and it's not a subtraction that, then it means it changes everything. So we're going to change this to a division because we realize that that is not the answer. Then if it's that, we're still going to keep the same because it's got brackets. We still do the bot mass. So this will be 18 x squared divided by 4 y divided by, we still need to simplify this. It goes 3 times and it goes 2 times. And therefore the answer here will be 2 over 3. Now, because it's division, remember we keep x squared over 4 y. We change the sign and we flip the equation. Now when we flip the equation, 2 can go into 18. 2 goes one time into 2 and it goes into 18. 9 times. Therefore 9 times 3 is 27. Should have been easier. X squared divided by 4 times 4 y times 1 is 4 y. Gosh, you know what I forgot is the x and the y. Sorry, my bad. There are x and y at the bottom. That's the other thing that we didn't take into consideration when we were simplifying this equation. Sorry. When we were simplifying this equation, we're going to have here at the bottom, it's going to be 3 y and 2 x. So therefore this will be 2 x and they will be y squared. And therefore it means that is not a division. Oh, yeah. I bet because in the original one that we had, I forgot to put the x and y. Hence I converted it to a division. Now I just realized that I'm now even thinking. But it's correct because it should be the same. So we can still simplify this. x squared and y and y over that. So y and y will cancel x and x squared. Remember x squared is the same as x times x. So one x will cancel out. Therefore the answer here will be 27 x over 4 because one of the x will cancel out. If you have x squared divided by x, we can also convert it and say because it's dividing. We can say x squared divided by x, they've got the same base, which is the same as x 2 minus 1, which is the same as x. That is the other way of solving it. So then it means you will be left with only x. And the same way y divided by y, which is y to the power of y, which is y divided by y to the power of y, which is y 1 minus 1, which is y to the power of 0, which is equals to 1, hence at the bottom. And y's disappeared. So the answer here will be option 2. So if it's true, this will be a division, not a subdirection. Okay, I'm going to give you this as your exercise. Simplify as far as possible. So yeah, we have 3 over x plus 4 over 2x plus 5 over 3x. Remember to go find your LCM first. I'm giving you a chance to do the activity on your own and then you will tell me what is the answer you get. And we will do it together. Are we winning? No question. Are you guys winning? No answer, no response. So I don't know whether you are getting it or you need help. No, you are done. Let's do it together. So because it's addition, we need to find the LCM. We have x, 2x, and 3x. So we'll have x, 2 times x, 3 times x, which in terms of LCM, it will be x times 2 times 3 times x. Oh, sorry. We cannot count x again because there is only one x on there. Sorry, my bad. So those will be the x's because we already counted x. So that x1 count, that x1 count because we already have an x in the first instance. So our LCM will be 2 times 3 with 6x. So that will be 6x. x goes how many times into 6x? It goes 6 times 6 times 3, 18. 2 goes how many times into 6? It goes 3 times. 3 times 4 is 12 because already x will cancel out. 3 goes how many times into 6? It goes 2 times, 2 times 5 is 10, and x will cancel out. 10 plus 12 is 18 plus 12 plus 10 is equals to 40. Divide 6x. What is the number that can go into 40 and also can go into 6 because we need to simplify this further. Who can go into both of them? Because 40 divided by 6, 6 cannot go into 40. So 2 goes how many times into 40? It goes 20 times and it goes how many times into 6? It goes 3 times and the answer will be option 4. Happy? Are you guys happy? The happiness in the house, are we all good? Yes, ma'am. We left with 15 minutes. Let's see if there is one more question. So this is just a fraction. Let's see if you still remember how to work it out. You have 3 over 8 plus 4 over 7 minus 9 over 14. If you have or if you own a cashier calculator, when it comes to fractions, you can use your fraction button to calculate in the exam. There is nothing wrong. You don't have to do it manually like we're doing it. The only manual thing you will do is when you are using variables. Otherwise, on your cashier calculator, you just put the equation as you see them using your fraction and it will calculate them the way and you will get the answer as quickly as possible. And it might be that this question also is one of those that has an error. The subtraction might be a division. Yeah, this is a division and you won't be able to find the answer if you don't use it as a division. And that is the problem with the past exam papers. In the exam, it will be clear because you get the original paper, not a scanned copy. Option 3. Okay, so how we got to option 3. Remember, board mass is the key to everything. 3 over 8 plus change, flip, and 7 goes one time and it goes two time. There is no other number that can divide into 9 and 4. So we will have the answer as 3 over 8 plus 4 times 2 is 8 over 1 times 9 is 9. Find the common denominator, it's 72. And that is some number that is very difficult to get to. Alternatively, you can say if you want to find the common denominator between two values, you can multiply those two values together, which is 8 times 9 is 72. So our common denominator will be 82. That is the easiest way of finding a common denominator. Like for example, sometimes it's easy, sometimes it's not easy. Like if you have 6, I'm just going to repeat the other one that we had, was it 6, 4, and 24. Now finding a common denominator between those three, it's easy because you can say it is 24 because 6 can go into 24 and 4 can go into 24 and 24 can go into itself. If you are going to multiply them, you will get a very big number. So you just need to be very careful when you use this method of multiplying the denominator, right? So I was just explaining that. But in this instance, we can use 8 times 9 will give us 72 because it's the LCM, which is the lowest common multiple between the two numbers. Okay, so 8 goes how many times into 72? It goes 9 times 9 times 3 is 27. Plus 9 goes how many times into 72? It goes 8 times and 8 times 8. It's 64 and 27 plus 64 is 91 divided by 72. 72 goes how many times into 91? It goes one time. The remainder will be 19 over 72. How do we get the remainder? You say 71 goes how many times into 92? Sorry, into 91 it goes one time. The remainder will be 91 minus 72. That's how you will find what the remainder is. The remainder will just be the subtraction of your numerator and the denominator. And that is 19 over 72, which is option number 3. So I've given you an additional question paper. Oh, sorry. In this question paper that we are looking at October, November, there are also some questions that we didn't cover. Like not this one, where it has the equation, we will cover next week. So not that. So actually we've covered everything. So all the other questions on this one, we will cover next week and the following weeks. So we've given you the other question paper, which is this one. You can go and practice. If you need any help, there is a WhatsApp group that I've created that you can also use to share ideas and share your calculation and find assistance from other people. But always remember when you do that, when you share your questions or your options, do not just share your options. Show us how you solved it. Take a picture of how you worked it out. Like I've been working out. So you will work it out the same way as you work it out and you take this picture, you email it to us so that we can tell you, oh, you went wrong here. Just fix that. Or you missed this step. Then you are able to know where your challenges are. So don't just send us the... I got the answer and answer is option three. We are not going to answer that for you. So we're just going to look at your question and leave it as it is. So you need to send us your work things as well. Okay, so the other thing, what we didn't cover because on the other question papers, they didn't have this type of things. It's if you get questions like this or questions that say, simplify this fraction but do not use a calculator or something like that. Find a decimal, change this to the C. It's easy to do. You should be able to answer that. So this is one of the LCN questions that you should have perceived in your assignment or when you were busy with your assignment or when you were practicing. But normally they don't ask this type of questions in your exam, but they do. They will ask you. So you'll never know because the lecture can change the way they ask questions. So if you get a question like this, you know how to solve it.  And just write the multiples. So what is the multiple of eight? It's two times two times two. Make sure that you get the lowest multiples. The multiple of X is X and X and the multiples of Y squared is Y and Y. You do the same here. 14 is two times seven times X times Y times Y. I don't have to put multiply by X, multiply by X because X and Y, X and X is the same as X multiplied by X. You don't have to put anything in between. Multiples of 12, so 12 can be two times six and six can, so it can be two times six and six, it's got its own multiples, which are two times three. So this will be two times two times three, which will give us the multiple of 12. So always start with the highest one and then see if you can find the multiples of the lowest one and then X times Y times Y times Y. So let's go find our LCM. Finding the LCM, we're going to keep all the values that we have right here at the top. The first line always will keep everything. So that will be two times two times two times X times X times Y times Y. Now when we go to the next one, anything that we already have, we don't have to include. But if it appears two times, we have to include the other one. So two, we already covered two into this. So we don't have to write two again. Seven, multiply by seven. X, we already covered the first X. Y, we already have Y. We already have the second Y. So we don't have to include anything else. Then we come to the next line. Two, we already have the first two. Two again, we already have the second two. Multiply by three, which we don't have. X, we already have the first X. And Y, we already have Y. And Y, we already have Y. And the remainder of the others would be multiplied by Y. So now we solve the numbers first. Two times two times two is eight times seven. So we'll say eight times seven times three equals to 168. And I don't even have to go and do the others in the exam. Or you can just go and circle the one that you have. And then let's do the X times X. We've got only those two X. So that will be X squared. How many Y's do we have? One, two, three. That will be Y cubed. And you can see that is the answer. So that is how you will answer the questions in the exam. And that concludes today's exam prep. I know that the session is two hours. And I really want to apologize for us starting late. But at least we have covered almost most of the concepts that we planned out to cover for today. And we did a whole lot of activities and exercises. Which should also give you some ideas in terms of preparations for your exam. So you can go and find more other activities to do. Go back to your past, your tutorial letters again. And look at those questions that they gave you initially. If you struggle to find the answer, try them again. If you still can get the answer right, we are here to support you. You can just send an email or use the WhatsApp group. And we should be able to help you with anything you need in order for you to prepare for the exam. Other than that, are there any questions? I didn't cover everything, right? So there are some questions here. So you will need to solve those fractions to get the answer. And I think that is the only fraction question that you have in this exam paper. And you can also go and find more question papers. You know, UNICEF students are very resourceful. There are question papers everywhere. So those are the ones that I have access to. If you haven't completed the register, please make sure that you do complete the register. Before we go, remember next week, we go into cover expression. Let me go to the content. You did receive the email, so you know which one we are covering. I think it's expression equations and any other equations that we have. Those who are doing QMI as well, you must also check what other sessions are we doing on QMI because we're not going to cover everything, everything, everything. We have limited time and limited hours as well. But I will try by all means to make sure that the difficult concepts that you are struggling with, you are able to go through and solve. So the next session that I have after this one is the QMI session. And I think it starts at 12 o'clock. Let me just give you a brief idea in terms of the type of... But then it will not benefit you because the concepts that we are discussing there are totally different to what you guys deal with. Other than that, those who are doing QMI, I will see you at 12 o'clock. Are there questions? Are there questions? I cannot add anyone on the group. You need to add yourself. I will send you now on here. Just stay on there. I will send you the link to the WhatsApp group. I cannot add you because if I add you, then it means I must add you on my phone and I'm not going to contact you anytime soon, which is pointless for me. So you can join the group by yourself. WhatsApp, right? That's where we need to be at. If there are no questions or comments other than the WhatsApp group, then we are done with today's session. But guys, someone needs to say something. Let's reflect. Was the session useful? Helpful? Can I hear some feedback? I found it really helpful. Thank you. Me too, ma'am. Thank you. Thank you. Next time when I ask, is there a question? I expect someone to at least say some comments. And usually I... I don't know whether you guys understand the concepts. You understand what we went through because when I ask a question and you are all quiet, you don't respond. You don't tell me we're still working. We are still busy or we are stuck. So it's a little bit frustrating, but I really appreciate some feedback and some engagement and some... I get the energy from you guys. But if you guys are quiet, I'm also going to be like, the session is going to drag along. And I will feel exhausted at the end of the session. But if you guys engage with me, talk to me, discuss with me. Ma'am. Sorry, ma'am. The reason I think why we're quiet is because when we filled out the form or something, it says that you agree to mute yourself on the calls. So I think that is why some of us are quiet because we don't know if we can unmute ourselves and talk. But when I ask if there are questions, then it means I expect you to unmute. At the beginning of the session, we ask you to mute yourself during the session and when we're doing some explanation and all that. So that then there is no noise in the background. But when it's time for engagement, when I ask is there a question or any comments, I expect you to unmute and talk to me. Because that is the opportunity that I'm giving you to talk to me now. Yes, ma'am. So next week is going to be a hyped session, right? Okay, ma'am. So you prefer us to unmute ourselves rather than put it in the chat? Yes, I prefer you to talk to me. Okay, ma'am.  And then, yes, only when we engage. Because now what happens when you place it in the chat, I don't see it until I go back to the chat because I don't see it. But if you engage with me and talk to me, it also makes the session interesting when we publish the session for the recording as well. Because now it's like you guys are not doing the work. I'm doing the work, right? Whereas I'm reading what you guys are giving me. And it's always nice to hear another voice than this husky voice that is on there. So I really appreciate that. You will unmute yourself next time I ask, what is your answer, any comments and all that. Okay, just a little bit. All right, I've got the link. I've just posted the link there on... This is BNU, right? Yes, BNU. You can join the WhatsApp group. And I will see you next week. Enjoy the rest of the weekend. Thank you, Mam, you too. All right, bye. Bye-bye.