Chapter 10. This is the Modern World
10.16 Sport and Leisure in Post-Confederation Canada
Robin Anderson; Department of History; and University of the Fraser Valley
The story of sport and leisure since the mid-19th century has been shaped by many factors, but three historical processes stand out in particular. The first is social hegemony in sport, the process by which dominant groups define the possibilities and limits on sport, leaving outlier groups having to struggle for inclusion. The second is fan identification and representation, the process in which a sport grows from a haphazard set of localized experiences to a phenomenon that fuels powerful allegiances to city, region, and nation. The third is commodification, the process of turning what originally was an informal and voluntary set of sport and leisure practices into a thoroughly commodified cultural marketplace. We will briefly look at each of these historical processes next.
Social Hegemony in Sport
As with most cultural activities, sport and leisure in the 19th and 20th centuries reflected the wider patterns of existence, social hierarchy, and class behaviours of Canadians. While human societies have always displayed a desire for play, organized sport with its emphasis on rational outcomes and fair play began in Great Britain in the late 18th century and was exported to British North America in the early 19th century as an extension of the leisure activities of the newly arrived British middle-class. Most historians agree that Montreal’s Anglophone middle-class men defined many of the enduring structures and practices of organized sport across Canada by the middle decades of the 19th century, including drafting the first rules of play, establishing the standard playing surfaces and equipment, creating the first sport organizations, leagues and events, and articulating a set of middle-class beliefs about the social and personal value of organized sport and leisure. Their vision of appropriate sport was a limited one but one that lined up closely in its broad outlines with similar developments in Europe and the United States.
While this relatively small group of sports-minded men had the money, leisure time, and social influence to define much of the physical activity in the 19th century, less powerful groups were still able to play their games within the nooks and crannies of rural and urban Canada. For instance, many of Canada’s Indigenous peoples continued to play their traditional games, although some of these, like the richly complex and multipurpose Iroquoian stick and ball game Tewaarathon, came under a re-forming process in the mid-19th century to become the standardized and rationalized sport lacrosse to mimic other contained European field games. Ironically, many eastern-Canadian Indigenous communities embraced the simplified European game of lacrosse and used their superior fitness and skills to regularly crush their white opponents. Other non-Anglo groups, including eastern European and Asian immigrants in the late 19th century, faced similar challenges to maintain their sport and leisure practices in the face of the dominant sport culture. And, while middle-class male enthusiasts refined the game of ice hockey, informal varieties of shinny (ice hockey) continued to be played on frozen ponds by rural and working class Canadians.
Organized sport was also designed in the 19th century as something men did, and was defined as a training ground for appropriate manhood. The masculinizing aims of sport shifted over time to reflect changes in the dominant form of manhood, but the attachment of sport to men has always been its central feature. This had, and still has, the effect of alienating some men who did not identify with the cultural messages around sport. And in much greater numbers, women were overtly excluded from most organized sports. By the late 19th century, however, women began to infiltrate and eventually demand equal access and resources within sport. The 1920s was a kind of golden decade for female athletes; high-profile Olympians Fanny “Bobbie” Rosenfeld (1904-69), Myrtle Cook (1902-85), and others simply represented the pinnacle of what were thousands of active women sports participants across the country. But the struggle was not over. Women’s efforts to gain an equal foothold in sport would continue through the 20th century in the face of ideological barriers about appropriate femininity, the supposed limitations on the female body, and overt male resistance. We live in a world today where sport opportunities are still heavily tipped in men’s favour and where media sport culture functions like a virtual man cave.
Fan Identification and Representation
Notwithstanding the large crowds for sporting events in ancient Greece and Rome, there was a time more recently when sport fans did not exist — and no one thought they should. Modern organized sport began in the early 19th century as private activities for the benefit of the participants alone. Over time, however, that changed. Sport spectators, known as “fans” today, emerged as an important feature of sporting life and they came to identify local sports teams and high profile athletes as representing them and their community. Sports historians refer to this phenomenon as the representational value of sport, and its emergence parallels the growing commercialization and professionalization of sport, and team sports in particular. Of course, Canada was not isolated from sporting developments elsewhere. In Britain in the early 19th century, many of the enduring patterns of sport identity and representation were established at competing public schools like Eton and Rugby; in the United States, professional baseball quickly took on representational value, following the American Civil War in the 1860s.
By the early 20th century, the notion of “following your team” was firmly established in Canada. And the energy of this identification, combined with the pocketbooks of fans, helped to drive the growth of professional sports. In Vancouver before the 1920s, as in most Canadian cities, three professional team sports — field lacrosse, baseball, and ice hockey — competed for the attention and allegiance of local fans, albeit in more limited regional settings. It was not until after WWII that city-based sport teams began to compete in leagues (such as the Canadian Football League) that were truly national in scope. Even so, the NHL lost three Canadian teams between the wars, as the league expanded into the United States in search of larger markets, and only added one — the Vancouver Canucks — before 1979. By the end of the 20th century, team sport identification had become arguably the most powerful communal attachment and experience that Canadians have. Today, the proliferation of local team jerseys worn as flags to identify fan loyalty, the ubiquitous media saturation that accompanies the games, the huge amounts of capital — both real and emotional — that are expended to support local teams demonstrate the apparently boundless power and appeal of sport representation. Ironically, however, the majority of players on the teams we choose to represent us are not from our communities, and most leave once their playing days are over.
Arguably the most potent representational force in sport is its connection to nationalism. Sport nationalism was slow to develop, and had to wait for the emergence and maturity of international sports organizations in the early 20th century. The modern Olympic Games, which began in 1896, first gave Canadians a taste of sport nationalism during the interwar years, demonstrating that even amateur athletics had the power to capture sports’ representational value. The 1928 Olympic Games produced sports heroes like two-time gold medalist sprinter Percy Williams (1908-82) and the successful women’s track and field team, the so-called “Matchless Six.” After the Second World War, Canada’s Olympic dreams shifted to the Winter Games, seen as more naturally suited to Canada’s northern character, with the success of skater Barbara Ann Scott (1928-2012) in 1948 and skier Nancy Greene (b. 1943) in 1968. Canada would eventually host three Olympic Games, in Montreal (1976), Calgary (1988), and Vancouver (2010), all of which became sites for intense national fervour and the rather dangerous emotional game of placing the fate of the nation in the hands of a very few. Canadians were disappointed most of the time.
Even more than the Olympic Games, Canadian sport nationalism has focused on international hockey. For most of the 20th century, the amateur code that governed most international sports organizations kept Canada’s best professional hockey players out of international competition. For generations, the National Hockey League (NHL) operated in a North American bubble, but with the confidence that the best hockey players in the world were those Canadians who toiled in the NHL. That began to change with the integrative forces of globalization from the 1960s onwards. In 1972, the NHL allowed its players to participate in an eight-game series with the best players from the Soviet Union. The resulting Summit Series and all the drama that the heroic narrative provided, capped by the iconic last-minute goal by Paul Henderson (b. 1943), became the high point of Canadian sports nationalism and, in the minds of Canadians for several decades, the most important national event in the country’s history. The importance of the Summit Series has begun to fade in the collective memory, but hockey nationalism certainly has not. An estimated 16 million Canadians crowded around their television sets or joined strangers in sport bars to watch both the Canadian women’s and men’s hockey teams win the 2010 Olympic gold medal in front of home audiences in Vancouver.
Before Canadians hand their self-identity and emotional welfare over to the performance of hockey players, it would be useful to remember several important points about hockey nationalism. First, no other country in the world regards ice hockey as its most important national sport. In the United States, it is at best the fourth most popular team sport, and several other professional sports such as golf and NASCAR rate much higher in attendance and viewership. When Canada won the two gold medals in Vancouver in 2010, both against American teams, the Americans graciously complimented the Canadians and then went about their business. In some European countries, like Sweden, Finland, and Russia, hockey occupies an important place, but lags far behind football (soccer) and other sports specific to their own sports heritage. Canada has chosen a sport to distinguish itself on the world stage that no one else cares about, at least not nearly as much as Canadians do. Canadian accomplishments on the soccer, rugby, or even cricket pitch would carry far more prestige on the world’s sporting stage.
We should also remember that a sport is an unusual and troubled way to define national character and purpose. Canada is one of the few countries that holds up a game as a definer of what that country is. Some cultural historians and critics have argued that Canada’s lack of coherent unity or any sense of collective values has forced it to embrace an activity as a symbol of its essential nature. Some suggest Canadians are just lazy and wish to avoid the conflicts that would accompany making those important choices. So, instead of choosing specific meaningful ideals that reflect and shape what Canada is as a country — a task that other countries have done — Canadians assign importance to the vagaries of a competitive game that presents conflicting messages to differing audiences, one that is played by a small and diminishing minority of mostly boys and men, and a game whose elite performers and major decision makers choose to work in the United States.
Commodification
The final factor that shaped the evolution of sport and recreation — commodification — is arguably the most important, and was probably unavoidable. The history of sport, after all, did not happen in a bubble, but rather paralleled the evolution of modern capitalism. As such, it has reflected the contours of social and cultural change that accompanied the industrial revolution. So, the initial emergence of organized sport forms in the mid-19th century has everything to do with the rise of industrial enterprise, the growth of wage labour, the expansion and increasing demands of the marketplace, the rapid growth of urban areas, the proliferation of industrial technology (especially advances in transportation), the invention of “leisure time,” and the growing confidence and recreational desires of the middle-class. When we talk about sport commodification, we mean the commercialization of events, which happened early, and the professionalization of players and athletes, which happened later and in some circles with greater controversy.
Right from the start, sport was attached to commercial opportunities, as enterprising individuals used athletic and physical activities to generate money. One-time sporting events, including prize-fights, canoe races, impromptu foot races, and strong-man competitions, were a common feature of 19th century Canadian life, and were almost always used as opportunities for gambling. One odd but typical example was the so-called “fat man” races that swept through Western Canada and the United States in the 1860s and 1870s. An advertisement in the Victoria British Colonist newspaper from 1872 gave readers ample notice:
Novel Race: A match has been made between two well-known Fat Men of this town, to run a foot race on the Queen’s birthday for 30 dollars a side. Beacon Hill [Park] will shake to its very base when these two great bodies are set in motion. Timid persons are requested to leave town for the day.[1]
Gambling helped fuel the process of commercialization in many sports in the 19th century. And with this economic energy came sports professionalism, first with paid athletes in individual sports, and then, by the end of the century, with professional rosters in team sports. The best known and certainly most colourful professional athlete in 19th century Canada was the world single-sculls rowing champion Ned Hanlan (1855-1908). Hanlan grew up in a working-class Irish family in the Toronto area and learned to row while helping his father illegally smuggle rum across Lake Ontario. After he mastered the new technology of the single-sculls sliding seat, a skill that was crucial to his singular success, Hanlan went on to dominate the enormously popular rowing competitions in Canada, the United States, and Britain in the 1880s. Hanlan and his backers knew how to milk the entertainment value and gambling opportunities of a close and uncertain outcome, and quite often the consummate showman Hanlan intentionally slowed down to make races appear closer than they actually were.
Sport historians are quick to remind us that even the strictest adherents of amateurism in the late 19th and early 20th centuries could not escape the needs of the marketplace, and even encouraged them. For instance, the Montreal Amateur Athletic Association (MAAA), the first and most powerful multisport organization in Canada before WWI, was formed in 1881 by Montreal middle-class sportsmen to pool capital to purchase land and club facilities. If they wanted to stay afloat, the MAAA and other amateur organizations were going to have to come to terms with the cash nexus. Money was needed for facility rentals, park development and the purchase of clubhouses, event expenses (including promotion and advertising), participant rewards and prizes, and the cost of players’ equipment and team accessories. All sports organizations — amateur included — charged gate admission to recover some or all of their related costs.
Amateur organizations like the MAAA were highly critical of professionalized team sports, which they saw as debasing the true spirit of voluntary recreation and the ideals of sport for sport’s sake. The potent forces that came to shape team sports, particularly the representational value of sport, 19th-century civic boosterism, and the overwhelming need to win, led to the practice of hiring the best players. Most major team sports moved incrementally from having the occasional paid player to stocking teams with dedicated professionals. By the early 20th century, most team sports were ignoring the voices of amateurism as they counted paying spectators filing into the ball parks, playing fields, and ice arenas across Canada.
One early example of the possibilities of commodification in team sports was the Pacific Coast Hockey Association (PCHA)F, a professional hockey operation that began in southwestern British Columbia by Frank and Lester Patrick in 1911. The Patrick brothers are remembered as creative innovators in hockey history, bringing in a slew of rule changes and practices that revolutionized the game, including the forward pass, the blue line, the use of in-play line changes, and the post-season playoff format. However, the PCHA was also a prime example of a fully commodified sports league. The Patricks owned and controlled every team in the PCHA, built state-of-the-art arenas with artificial ice, and moved teams around frequently to maximize revenue. They stocked the league with star players that were lured (some say raided) from the National Hockey Association (NHA), the leading professional hockey league in eastern Canada that would become the National Hockey League (NHL) in 1917. The best known of these star players was Edouard “Newsey” Lalonde (1887-1970) and Fred “Cyclone” Taylor (1884-1979). Centralized team ownership meant that the Patricks could control player salaries and manipulate team rosters to achieve parity in the league, two issues that vex professional leagues today. The stability and public profile of the PCHA forced the NHA into a predictable east-west playoff structure for the Stanley Cup, effectively ending the challenge cup format that was in place since 1893. As a result of the Patricks’ investments, the Vancouver Millionaires were Stanley Cup champions in 1915, winning a best of three series from the Ottawa Senators three games to none. The PCHA’s Seattle Metropolitans would also win the Cup in 1917.
The opportunities for profit in sport increased exponentially in the second half of the 20th century. From the 1960s onwards, Canadians experienced an acceleration of commodification as the marketplace entered every area of life. Sport was part of this process. The revenue streams available to professional sports began to increase with television coverage in the 1960s and the delivery of commercial advertising through sports entertainment. Rising ticket prices to games reflected the higher income levels of certain segments of the Canadian population. Player salaries also rose and soon escalated to keep step with the growing profits available in professional sports leagues. By the 1980s, sport commodification moved beyond television contracts and ticket revenues into retail sales and sponsorship agreements. Today, professional sport is a foundational element of modern capitalism because its emotional appeal is so easily attached to the capitalist marketplace.
The forces of sport commodification would eventually kill amateurism. The Olympic Games, or the “Olympic movement” as advocates prefer to call it, positioned itself early in the 20th century as the global defender of amateur principles. However, the lure of increased revenues for the Games after WWII proved too enticing even for members of the International Olympic Committee (IOC), the appointed officials who governed the organization. What began as relatively small television contracts for the Games in the mid-1950s soon mushroomed in size to reflect profits to be made in media coverage. The IOC added sponsorship agreements to its revenue pool in the 1980s, as giant corporations such as McDonald’s, Coca-Cola, and Visa payed millions of dollars to be official partners for the Games. Once the commercial door was blown open by huge amounts of corporate money, it proved difficult to keep professionals out of the Games and maintain the principles of amateurism. For one, the public demanded to see the best athletes perform, professional or not. To accommodate, in 1988 the IOC allowed the international sports federations that governed each sport to define Olympic participation themselves. Most would eventually choose to abandon the amateur code.
Today, the Olympic Games stand as the quintessential example of a thoroughly commodified sports entity. Its revenues rival the annual income of some countries, its commercial tentacles penetrate every market opportunity imaginable, and its boardroom of lawyers vigilantly polices its trademarks and other marketable symbols. But of course it is not alone. We uncritically admit today that modern sports organizations are businesses, first and foremost, and in so doing we allow them whatever latitude available to pursue their economic goals. However, we usually fail to recognize the underlying context of expansionary capitalism that has moved sport and sport entertainment into the centre of our lives. We also fail to see the gap that has developed between elite professional sports celebrities and the rank-and-file sports fan.
Key Points
- Historians of sport observe that social hegemony works to privilege some activities and marginalize others, which in Canada has meant the middle class definition of appropriate organized games was a racialized and sexualized — white and male — space.
- Spectator sports are historically tied to the notion of the representational value of the games, which is tied to urbanization and the professionalization of playing.
- Canada’s obsession with ice hockey has eclipsed interest in sports with a far greater international cachet.
- Industrialization and urbanization contributed to the emergence of leisure time and the use of disposable income to support sports enterprises. Sports became professionalized and turned into a commodity for sale.
Additional Readings
Bouchier, Nancy. For the Love of the Game: Amateur Sport in Small-Town Ontario. Montreal: McGill-Queen’s University Press, 2003.
Burstyn, Varda. The Rites of Men: Manhood, Politics and the Culture of Sport. Toronto: University of Toronto Press, 1999.
Hall, M. Ann. The Girl and the Game: A History of Women’s Sport in Canada. Toronto: University of Toronto Press, 2002.
Gruneau, Richard and Whitson, David. Hockey Night in Canada: Sports, Identities, and Cultural Politics. Toronto: University of Toronto Press, 1994.
Howell, Colin. Blood, Sweat, and Cheers: Sport and the Making of Modern Canada. Toronto: University of Toronto Press, 2001.
Kidd, Bruce. The Struggle for Canadian Sport. Toronto: University of Toronto Press, 1996.
Morrow, Don and Wamsley, Kevin B. Sport in Canada: A History. 3rd Edition. Toronto: Oxford University Press, 2013.
Poulter, Gillian. “Montreal and its Environs: Imaging a National Landscape, c. 1867-1885” Journal of Canadian Studies 38:3 (2004): 69-100.
Media Attributions
- Hockey, 1919 © H.J. Woodside, Library and Archives Canada (C-079313) is licensed under a Public Domain license
- Caughnawaga Indian Lacrosse players: study of five figures © Library and Archives Canada, Acc. No. R9266-224. Peter Winkworth Collection of Canadiana is licensed under a Public Domain license
- Myrtle Cook of Canada (left) winning a preliminary heat in the women’s 100 metres race at the VIIIth Summer Olympic Games © Library and Archives Canada (PA-150994) is licensed under a Public Domain license
- Fernie Swastikas hockey team, 1922 © Fernie Swastikas is licensed under a Public Domain license
- Clint Benedict, Montreal Maroons © Hockey Hall of Fame, Library and Archives Canada (PA-048898) is licensed under a Public Domain license
- Cricket and baseball – Willowvale Park © Toronto City Archives, Fonds 1548, Series 393, Item 16432 is licensed under a Public Domain license
- Edward (Ned) Hanlan © Library and Archives Canada, Acc. No. R9266-3343. Peter Winkworth Collection of Canadiana is licensed under a Public Domain license
- 1914_Vancouver_Millionaires © Stuart Thomson, City of Vancouver Archives (CVA 99-126) is licensed under a Public Domain license
- Highland Games, Antigonish, Aug. 1940, athlete throwing stone © Ronny Jacques, Library and Archives Canada (MIKAN no. 4315254) is licensed under a Public Domain license
- Ad from the British Colonist (Victoria, 1861), cited in Gary Kingston, "B.C.'s first "sports event" probably a cricket match", Vancouver Sun, (10 November 2009), accessed 13 May 2016, http://www.vancouversun.com/first+sports+event+probably+cricket+match/2207952/story.html. ↵
Influence enjoyed by one social group over all others; dominance in tastes, culture, and values, among other indicators.
In the context of, principally, professional sports, the phenomenon of fan allegiance to a team or player; manifest in the wearing of sports merchandise or loyalty to a team or club, and consciously encouraged by local media.
In the context of the professionalization of sports and leisure, the process of turning what originally was an informal and voluntary set of practices into a commodity to be bought and sold.
In the context of, principally, professional sports, the phenomenon of athletes whose performance is seen by the community and by fans especially as representing the community and its members; applies to local and to national players/teams.
Generally, the creation of exclusive policies that limit entry into a particular business or trade, such as the need for a teaching certificate from a recognized institution in order to become a teacher; in sports, the phenomenon of paying players to play, which moved games and athleticism away from the 19th century ideal of gentlemanly and unpaid (amateur) competition.
In the context of the history of modern sport, refers to athletes who do not accept pay to play; also implies a middle- and upper-middle class ethos of fairplay and a hostility toward professionalism.
Established in 1917 after a dispute among team owners in the National Hockey Association. It was, originally, an all-Canadian league but expanded in 1920 to Boston. Its higher salaries and American market led to the decline and disappearance of other professional leagues and the rise of an effective monopoly by the 1940s.
Created in 1881, a federation of non-professional sports organizations, including bicycling, lacrosse, and ice hockey clubs; argued for a gentlemanly view of athletics, one which built character and community; opposed to the professionalization of sports and games.
One of several early 20th century professional hockey leagues; pioneered use of artificial ice in indoor arenas; merged with the Western Canada Hockey League in 1924.
One of several early 20th century professional hockey leagues and the direct precursor of the National Hockey League.
Established 1894; responsible for the organization and operation of the Olympic Games (both winter and summer versions).


Beyond Good and Evil
Chapter 6: We Scholars
205. The dangers that beset the evolution of the philosopher are, in fact, so manifold nowadays, that one might doubt whether this fruit could still come to maturity. The extent and towering structure of the sciences have increased enormously, and therewith also the probability that the philosopher will grow tired even as a learner, or will attach himself somewhere and “specialize” so that he will no longer attain to his elevation, that is to say, to his superspection, his circumspection, and his despection. Or he gets aloft too late, when the best of his maturity and strength is past, or when he is impaired, coarsened, and deteriorated, so that his view, his general estimate of things, is no longer of much importance. It is perhaps just the refinement of his intellectual conscience that makes him hesitate and linger on the way, he dreads the temptation to become a dilettante, a millepede, a milleantenna, he knows too well that as a discerner, one who has lost his self-respect no longer commands, no longer leads, unless he should aspire to become a great play-actor, a philosophical Cagliostro and spiritual rat-catcher—in short, a misleader. This is in the last instance a question of taste, if it has not really been a question of conscience. To double once more the philosopher’s difficulties, there is also the fact that he demands from himself a verdict, a Yea or Nay, not concerning science, but concerning life and the worth of life—he learns unwillingly to believe that it is his right and even his duty to obtain this verdict, and he has to seek his way to the right and the belief only through the most extensive (perhaps disturbing and destroying) experiences, often hesitating, doubting, and dumbfounded. In fact, the philosopher has long been mistaken and confused by the multitude, either with the scientific man and ideal scholar, or with the religiously elevated, desensualized, desecularized visionary and God-intoxicated man; and even yet when one hears anybody praised, because he lives “wisely,” or “as a philosopher,” it hardly means anything more than “prudently and apart.” Wisdom: that seems to the populace to be a kind of flight, a means and artifice for withdrawing successfully from a bad game; but the genuine philosopher—does it not seem so to us, my friends?—lives “unphilosophically” and “unwisely,” above all, imprudently, and feels the obligation and burden of a hundred attempts and temptations of life—he risks himself constantly, he plays this bad game.
206. In relation to the genius, that is to say, a being who either engenders or produces—both words understood in their fullest sense—the man of learning, the scientific average man, has always something of the old maid about him; for, like her, he is not conversant with the two principal functions of man. To both, of course, to the scholar and to the old maid, one concedes respectability, as if by way of indemnification—in these cases one emphasizes the respectability—and yet, in the compulsion of this concession, one has the same admixture of vexation. Let us examine more closely: what is the scientific man? Firstly, a commonplace type of man, with commonplace virtues: that is to say, a non-ruling, non-authoritative, and non-self-sufficient type of man; he possesses industry, patient adaptableness to rank and file, equability and moderation in capacity and requirement; he has the instinct for people like himself, and for that which they require—for instance: the portion of independence and green meadow without which there is no rest from labour, the claim to honour and consideration (which first and foremost presupposes recognition and recognisability), the sunshine of a good name, the perpetual ratification of his value and usefulness, with which the inward distrust which lies at the bottom of the heart of all dependent men and gregarious animals, has again and again to be overcome. The learned man, as is appropriate, has also maladies and faults of an ignoble kind: he is full of petty envy, and has a lynx-eye for the weak points in those natures to whose elevations he cannot attain. He is confiding, yet only as one who lets himself go, but does not flow; and precisely before the man of the great current he stands all the colder and more reserved—his eye is then like a smooth and irresponsive lake, which is no longer moved by rapture or sympathy. The worst and most dangerous thing of which a scholar is capable results from the instinct of mediocrity of his type, from the Jesuitism of mediocrity, which labours instinctively for the destruction of the exceptional man, and endeavours to break—or still better, to relax—every bent bow To relax, of course, with consideration, and naturally with an indulgent hand—to relax with confiding sympathy that is the real art of Jesuitism, which has always understood how to introduce itself as the religion of sympathy.
207. However gratefully one may welcome the objective spirit—and who has not been sick to death of all subjectivity and its confounded ipsisimosity!—in the end, however, one must learn caution even with regard to one’s gratitude, and put a stop to the exaggeration with which the unselfing and depersonalizing of the spirit has recently been celebrated, as if it were the goal in itself, as if it were salvation and glorification—as is especially accustomed to happen in the pessimist school, which has also in its turn good reasons for paying the highest honours to “disinterested knowledge” The objective man, who no longer curses and scolds like the pessimist, the ideal man of learning in whom the scientific instinct blossoms forth fully after a thousand complete and partial failures, is assuredly one of the most costly instruments that exist, but his place is in the hand of one who is more powerful He is only an instrument, we may say, he is a mirror—he is no “purpose in himself” The objective man is in truth a mirror accustomed to prostration before everything that wants to be known, with such desires only as knowing or “reflecting” implies—he waits until something comes, and then expands himself sensitively, so that even the light footsteps and gliding-past of spiritual beings may not be lost on his surface and film Whatever “personality” he still possesses seems to him accidental, arbitrary, or still oftener, disturbing, so much has he come to regard himself as the passage and reflection of outside forms and events He calls up the recollection of “himself” with an effort, and not infrequently wrongly, he readily confounds himself with other persons, he makes mistakes with regard to his own needs, and here only is he unrefined and negligent Perhaps he is troubled about the health, or the pettiness and confined atmosphere of wife and friend, or the lack of companions and society—indeed, he sets himself to reflect on his suffering, but in vain! His thoughts already rove away to the more general case, and tomorrow he knows as little as he knew yesterday how to help himself He does not now take himself seriously and devote time to himself he is serene, not from lack of trouble, but from lack of capacity for grasping and dealing with his trouble The habitual complaisance with respect to all objects and experiences, the radiant and impartial hospitality with which he receives everything that comes his way, his habit of inconsiderate good-nature, of dangerous indifference as to Yea and Nay: alas! there are enough of cases in which he has to atone for these virtues of his!—and as man generally, he becomes far too easily the caput mortuum of such virtues. Should one wish love or hatred from him—I mean love and hatred as God, woman, and animal understand them—he will do what he can, and furnish what he can. But one must not be surprised if it should not be much—if he should show himself just at this point to be false, fragile, questionable, and deteriorated. His love is constrained, his hatred is artificial, and rather un tour de force, a slight ostentation and exaggeration. He is only genuine so far as he can be objective; only in his serene totality is he still “nature” and “natural.” His mirroring and eternally self-polishing soul no longer knows how to affirm, no longer how to deny; he does not command; neither does he destroy. “Je ne meprise presque rien“—he says, with Leibniz: let us not overlook nor undervalue the presque! Neither is he a model man; he does not go in advance of any one, nor after, either; he places himself generally too far off to have any reason for espousing the cause of either good or evil. If he has been so long confounded with the philosopher, with the Caesarian trainer and dictator of civilization, he has had far too much honour, and what is more essential in him has been overlooked—he is an instrument, something of a slave, though certainly the sublimest sort of slave, but nothing in himself—presque rien! The objective man is an instrument, a costly, easily injured, easily tarnished measuring instrument and mirroring apparatus, which is to be taken care of and respected; but he is no goal, not outgoing nor upgoing, no complementary man in whom the rest of existence justifies itself, no termination—and still less a commencement, an engendering, or primary cause, nothing hardy, powerful, self-centred, that wants to be master; but rather only a soft, inflated, delicate, movable potter’s-form, that must wait for some kind of content and frame to “shape” itself thereto—for the most part a man without frame and content, a “selfless” man. Consequently, also, nothing for women, in parenthesi.
208. When a philosopher nowadays makes known that he is not a skeptic—I hope that has been gathered from the foregoing description of the objective spirit?—people all hear it impatiently; they regard him on that account with some apprehension, they would like to ask so many, many questions… indeed among timid hearers, of whom there are now so many, he is henceforth said to be dangerous. With his repudiation of skepticism, it seems to them as if they heard some evil-threatening sound in the distance, as if a new kind of explosive were being tried somewhere, a dynamite of the spirit, perhaps a newly discovered Russian nihiline, a pessimism bone voluntatis, that not only denies, means denial, but—dreadful thought! Practises denial. Against this kind of “good-will”—a will to the veritable, actual negation of life—there is, as is generally acknowledged nowadays, no better soporific and sedative than skepticism, the mild, pleasing, lulling poppy of skepticism; and Hamlet himself is now prescribed by the doctors of the day as an antidote to the “spirit,” and its underground noises. “Are not our ears already full of bad sounds?” say the skeptics, as lovers of repose, and almost as a kind of safety police; “this subterranean Nay is terrible! Be still, ye pessimistic moles!” The skeptic, in effect, that delicate creature, is far too easily frightened; his conscience is schooled so as to start at every Nay, and even at that sharp, decided Yea, and feels something like a bite thereby. Yea! and Nay!—they seem to him opposed to morality; he loves, on the contrary, to make a festival to his virtue by a noble aloofness, while perhaps he says with Montaigne: “What do I know?” Or with Socrates: “I know that I know nothing.” Or: “Here I do not trust myself, no door is open to me.” Or: “Even if the door were open, why should I enter immediately?” Or: “What is the use of any hasty hypotheses? It might quite well be in good taste to make no hypotheses at all. Are you absolutely obliged to straighten at once what is crooked? to stuff every hole with some kind of oakum? Is there not time enough for that? Has not the time leisure? Oh, ye demons, can ye not at all wait? The uncertain also has its charms, the Sphinx, too, is a Circe, and Circe, too, was a philosopher.”—Thus does a skeptic console himself; and in truth he needs some consolation. For skepticism is the most spiritual expression of a certain many-sided physiological temperament, which in ordinary language is called nervous debility and sickliness; it arises whenever races or classes which have been long separated, decisively and suddenly blend with one another. In the new generation, which has inherited as it were different standards and valuations in its blood, everything is disquiet, derangement, doubt, and tentativeness; the best powers operate restrictively, the very virtues prevent each other growing and becoming strong, equilibrium, ballast, and perpendicular stability are lacking in body and soul. That, however, which is most diseased and degenerated in such nondescripts is the WILL; they are no longer familiar with independence of decision, or the courageous feeling of pleasure in willing—they are doubtful of the “freedom of the will” even in their dreams Our present-day Europe, the scene of a senseless, precipitate attempt at a radical blending of classes, and consequently of races, is therefore skeptical in all its heights and depths, sometimes exhibiting the mobile skepticism which springs impatiently and wantonly from branch to branch, sometimes with gloomy aspect, like a cloud over-charged with interrogative signs—and often sick unto death of its will! Paralysis of will, where do we not find this cripple sitting nowadays! And yet how bedecked oftentimes’ How seductively ornamented! There are the finest gala dresses and disguises for this disease, and that, for instance, most of what places itself nowadays in the show-cases as “objectiveness,” “the scientific spirit,” “l’art pour l’art,” and “pure voluntary knowledge,” is only decked-out skepticism and paralysis of will—I am ready to answer for this diagnosis of the European disease—The disease of the will is diffused unequally over Europe, it is worst and most varied where civilization has longest prevailed, it decreases according as “the barbarian” still—or again—asserts his claims under the loose drapery of Western culture It is therefore in the France of today, as can be readily disclosed and comprehended, that the will is most infirm, and France, which has always had a masterly aptitude for converting even the portentous crises of its spirit into something charming and seductive, now manifests emphatically its intellectual ascendancy over Europe, by being the school and exhibition of all the charms of skepticism The power to will and to persist, moreover, in a resolution, is already somewhat stronger in Germany, and again in the North of Germany it is stronger than in Central Germany, it is considerably stronger in England, Spain, and Corsica, associated with phlegm in the former and with hard skulls in the latter—not to mention Italy, which is too young yet to know what it wants, and must first show whether it can exercise will, but it is strongest and most surprising of all in that immense middle empire where Europe as it were flows back to Asia—namely, in Russia There the power to will has been long stored up and accumulated, there the will—uncertain whether to be negative or affirmative—waits threateningly to be discharged (to borrow their pet phrase from our physicists) Perhaps not only Indian wars and complications in Asia would be necessary to free Europe from its greatest danger, but also internal subversion, the shattering of the empire into small states, and above all the introduction of parliamentary imbecility, together with the obligation of every one to read his newspaper at breakfast I do not say this as one who desires it, in my heart I should rather prefer the contrary—I mean such an increase in the threatening attitude of Russia, that Europe would have to make up its mind to become equally threatening—namely, to acquire one will, by means of a new caste to rule over the Continent, a persistent, dreadful will of its own, that can set its aims thousands of years ahead; so that the long spun-out comedy of its petty-statism, and its dynastic as well as its democratic many-willed-ness, might finally be brought to a close. The time for petty politics is past; the next century will bring the struggle for the dominion of the world—the compulsion to great politics.
209. As to how far the new warlike age on which we Europeans have evidently entered may perhaps favour the growth of another and stronger kind of skepticism, I should like to express myself preliminarily merely by a parable, which the lovers of German history will already understand. That unscrupulous enthusiast for big, handsome grenadiers (who, as King of Prussia, brought into being a military and skeptical genius—and therewith, in reality, the new and now triumphantly emerged type of German), the problematic, crazy father of Frederick the Great, had on one point the very knack and lucky grasp of the genius: he knew what was then lacking in Germany, the want of which was a hundred times more alarming and serious than any lack of culture and social form—his ill-will to the young Frederick resulted from the anxiety of a profound instinct. Men were lacking; and he suspected, to his bitterest regret, that his own son was not man enough. There, however, he deceived himself; but who would not have deceived himself in his place? He saw his son lapsed to atheism, to the esprit, to the pleasant frivolity of clever Frenchmen—he saw in the background the great bloodsucker, the spider skepticism; he suspected the incurable wretchedness of a heart no longer hard enough either for evil or good, and of a broken will that no longer commands, is no longer able to command. Meanwhile, however, there grew up in his son that new kind of harder and more dangerous skepticism—who knows to what extent it was encouraged just by his father’s hatred and the icy melancholy of a will condemned to solitude?—the skepticism of daring manliness, which is closely related to the genius for war and conquest, and made its first entrance into Germany in the person of the great Frederick. This skepticism despises and nevertheless grasps; it undermines and takes possession; it does not believe, but it does not thereby lose itself; it gives the spirit a dangerous liberty, but it keeps strict guard over the heart. It is the German form of skepticism, which, as a continued Fredericianism, risen to the highest spirituality, has kept Europe for a considerable time under the dominion of the German spirit and its critical and historical distrust Owing to the insuperably strong and tough masculine character of the great German philologists and historical critics (who, rightly estimated, were also all of them artists of destruction and dissolution), a new conception of the German spirit gradually established itself—in spite of all Romanticism in music and philosophy—in which the leaning towards masculine skepticism was decidedly prominent whether, for instance, as fearlessness of gaze, as courage and sternness of the dissecting hand, or as resolute will to dangerous voyages of discovery, to spiritualized North Pole expeditions under barren and dangerous skies. There may be good grounds for it when warm-blooded and superficial humanitarians cross themselves before this spirit, cet esprit fataliste, ironique, mephistophelique, as Michelet calls it, not without a shudder. But if one would realize how characteristic is this fear of the “man” in the German spirit which awakened Europe out of its “dogmatic slumber,” let us call to mind the former conception which had to be overcome by this new one—and that it is not so very long ago that a masculinized woman could dare, with unbridled presumption, to recommend the Germans to the interest of Europe as gentle, good-hearted, weak-willed, and poetical fools. Finally, let us only understand profoundly enough Napoleon’s astonishment when he saw Goethe it reveals what had been regarded for centuries as the “German spirit” “voila un homme!”—that was as much as to say “But this is a man! And I only expected to see a German!”
210. Supposing, then, that in the picture of the philosophers of the future, some trait suggests the question whether they must not perhaps be skeptics in the last-mentioned sense, something in them would only be designated thereby—and not they themselves. With equal right they might call themselves critics, and assuredly they will be men of experiments. By the name with which I ventured to baptize them, I have already expressly emphasized their attempting and their love of attempting is this because, as critics in body and soul, they will love to make use of experiments in a new, and perhaps wider and more dangerous sense? In their passion for knowledge, will they have to go further in daring and painful attempts than the sensitive and pampered taste of a democratic century can approve of?—There is no doubt these coming ones will be least able to dispense with the serious and not unscrupulous qualities which distinguish the critic from the skeptic I mean the certainty as to standards of worth, the conscious employment of a unity of method, the wary courage, the standing-alone, and the capacity for self-responsibility, indeed, they will avow among themselves a delight in denial and dissection, and a certain considerate cruelty, which knows how to handle the knife surely and deftly, even when the heart bleeds They will be sterner (and perhaps not always towards themselves only) than humane people may desire, they will not deal with the “truth” in order that it may “please” them, or “elevate” and “inspire” them—they will rather have little faith in “truth” bringing with it such revels for the feelings. They will smile, those rigorous spirits, when any one says in their presence “That thought elevates me, why should it not be true?” or “That work enchants me, why should it not be beautiful?” or “That artist enlarges me, why should he not be great?” Perhaps they will not only have a smile, but a genuine disgust for all that is thus rapturous, idealistic, feminine, and hermaphroditic, and if any one could look into their inmost hearts, he would not easily find therein the intention to reconcile “Christian sentiments” with “antique taste,” or even with “modern parliamentarism” (the kind of reconciliation necessarily found even among philosophers in our very uncertain and consequently very conciliatory century). Critical discipline, and every habit that conduces to purity and rigour in intellectual matters, will not only be demanded from themselves by these philosophers of the future, they may even make a display thereof as their special adornment—nevertheless they will not want to be called critics on that account. It will seem to them no small indignity to philosophy to have it decreed, as is so welcome nowadays, that “philosophy itself is criticism and critical science—and nothing else whatever!” Though this estimate of philosophy may enjoy the approval of all the Positivists of France and Germany (and possibly it even flattered the heart and taste of kant: let us call to mind the titles of his principal works), our new philosophers will say, notwithstanding, that critics are instruments of the philosopher, and just on that account, as instruments, they are far from being philosophers themselves! Even the great Chinaman of Konigsberg was only a great critic.
211. I insist upon it that people finally cease confounding philosophical workers, and in general scientific men, with philosophers—that precisely here one should strictly give “each his own,” and not give those far too much, these far too little. It may be necessary for the education of the real philosopher that he himself should have once stood upon all those steps upon which his servants, the scientific workers of philosophy, remain standing, and must remain standing he himself must perhaps have been critic, and dogmatist, and historian, and besides, poet, and collector, and traveler, and riddle-reader, and moralist, and seer, and “free spirit,” and almost everything, in order to traverse the whole range of human values and estimations, and that he may be able with a variety of eyes and consciences to look from a height to any distance, from a depth up to any height, from a nook into any expanse. But all these are only preliminary conditions for his task; this task itself demands something else—it requires him to create values. The philosophical workers, after the excellent pattern of Kant and Hegel, have to fix and formalize some great existing body of valuations—that is to say, former determinations of value, creations of value, which have become prevalent, and are for a time called “truths”—whether in the domain of the logical, the political (moral), or the artistic. It is for these investigators to make whatever has happened and been esteemed hitherto, conspicuous, conceivable, intelligible, and manageable, to shorten everything long, even “time” itself, and to subjugate the entire past: an immense and wonderful task, in the carrying out of which all refined pride, all tenacious will, can surely find satisfaction. The real philosophers, however, are commanders and law-givers; they say: “Thus shall it be!” They determine first the Whither and the Why of mankind, and thereby set aside the previous labour of all philosophical workers, and all subjugators of the past—they grasp at the future with a creative hand, and whatever is and was, becomes for them thereby a means, an instrument, and a hammer. Their “knowing” is creating, their creating is a law-giving, their will to truth is—will to power.—Are there at present such philosophers? Have there ever been such philosophers? Must there not be such philosophers some day? …
212. It is always more obvious to me that the philosopher, as a man indispensable for the morrow and the day after the morrow, has ever found himself, and has been obliged to find himself, in contradiction to the day in which he lives; his enemy has always been the ideal of his day. Hitherto all those extraordinary furtherers of humanity whom one calls philosophers—who rarely regarded themselves as lovers of wisdom, but rather as disagreeable fools and dangerous interrogators—have found their mission, their hard, involuntary, imperative mission (in the end, however, the greatness of their mission), in being the bad conscience of their age. In putting the vivisector’s knife to the breast of the very virtues of their age, they have betrayed their own secret; it has been for the sake of a new greatness of man, a new untrodden path to his aggrandizement. They have always disclosed how much hypocrisy, indolence, self-indulgence, and self-neglect, how much falsehood was concealed under the most venerated types of contemporary morality, how much virtue was outlived, they have always said “We must remove hence to where you are least at home” In the face of a world of “modern ideas,” which would like to confine every one in a corner, in a “specialty,” a philosopher, if there could be philosophers nowadays, would be compelled to place the greatness of man, the conception of “greatness,” precisely in his comprehensiveness and multifariousness, in his all-roundness, he would even determine worth and rank according to the amount and variety of that which a man could bear and take upon himself, according to the extent to which a man could stretch his responsibility Nowadays the taste and virtue of the age weaken and attenuate the will, nothing is so adapted to the spirit of the age as weakness of will consequently, in the ideal of the philosopher, strength of will, sternness, and capacity for prolonged resolution, must specially be included in the conception of “greatness”, with as good a right as the opposite doctrine, with its ideal of a silly, renouncing, humble, selfless humanity, was suited to an opposite age—such as the sixteenth century, which suffered from its accumulated energy of will, and from the wildest torrents and floods of selfishness In the time of Socrates, among men only of worn-out instincts, old conservative Athenians who let themselves go—”for the sake of happiness,” as they said, for the sake of pleasure, as their conduct indicated—and who had continually on their lips the old pompous words to which they had long forfeited the right by the life they led, irony was perhaps necessary for greatness of soul, the wicked Socratic assurance of the old physician and plebeian, who cut ruthlessly into his own flesh, as into the flesh and heart of the “noble,” with a look that said plainly enough “Do not dissemble before me! here—we are equal!” At present, on the contrary, when throughout Europe the herding-animal alone attains to honours, and dispenses honours, when “equality of right” can too readily be transformed into equality in wrong—I mean to say into general war against everything rare, strange, and privileged, against the higher man, the higher soul, the higher duty, the higher responsibility, the creative plenipotence and lordliness—at present it belongs to the conception of “greatness” to be noble, to wish to be apart, to be capable of being different, to stand alone, to have to live by personal initiative, and the philosopher will betray something of his own ideal when he asserts “He shall be the greatest who can be the most solitary, the most concealed, the most divergent, the man beyond good and evil, the master of his virtues, and of super-abundance of will; precisely this shall be called greatness: as diversified as can be entire, as ample as can be full.” And to ask once more the question: Is greatness possible—nowadays?
213. It is difficult to learn what a philosopher is, because it cannot be taught: one must “know” it by experience—or one should have the pride not to know it. The fact that at present people all talk of things of which they cannot have any experience, is true more especially and unfortunately as concerns the philosopher and philosophical matters:—the very few know them, are permitted to know them, and all popular ideas about them are false. Thus, for instance, the truly philosophical combination of a bold, exuberant spirituality which runs at presto pace, and a dialectic rigour and necessity which makes no false step, is unknown to most thinkers and scholars from their own experience, and therefore, should any one speak of it in their presence, it is incredible to them. They conceive of every necessity as troublesome, as a painful compulsory obedience and state of constraint; thinking itself is regarded by them as something slow and hesitating, almost as a trouble, and often enough as “worthy of the sweat of the noble”—but not at all as something easy and divine, closely related to dancing and exuberance! “To think” and to take a matter “seriously,” “arduously”—that is one and the same thing to them; such only has been their “experience.”—Artists have here perhaps a finer intuition; they who know only too well that precisely when they no longer do anything “arbitrarily,” and everything of necessity, their feeling of freedom, of subtlety, of power, of creatively fixing, disposing, and shaping, reaches its climax—in short, that necessity and “freedom of will” are then the same thing with them. There is, in fine, a gradation of rank in psychical states, to which the gradation of rank in the problems corresponds; and the highest problems repel ruthlessly every one who ventures too near them, without being predestined for their solution by the loftiness and power of his spirituality. Of what use is it for nimble, everyday intellects, or clumsy, honest mechanics and empiricists to press, in their plebeian ambition, close to such problems, and as it were into this “holy of holies”—as so often happens nowadays! But coarse feet must never tread upon such carpets: this is provided for in the primary law of things; the doors remain closed to those intruders, though they may dash and break their heads thereon. People have always to be born to a high station, or, more definitely, they have to be bred for it: a person has only a right to philosophy—taking the word in its higher significance—in virtue of his descent; the ancestors, the “blood,” decide here also. Many generations must have prepared the way for the coming of the philosopher; each of his virtues must have been separately acquired, nurtured, transmitted, and embodied; not only the bold, easy, delicate course and current of his thoughts, but above all the readiness for great responsibilities, the majesty of ruling glance and contemning look, the feeling of separation from the multitude with their duties and virtues, the kindly patronage and defense of whatever is misunderstood and calumniated, be it God or devil, the delight and practice of supreme justice, the art of commanding, the amplitude of will, the lingering eye which rarely admires, rarely looks up, rarely loves….


28 Combining CC Licenses
Combining Different CC Licenses |
Can I combine works that use different Creative Commons licenses into my work?
- It depends. You may only remix works released under different CC licenses if the terms of the licenses involved permit it. The NoDerivatives licenses do not permit remixing at all. All the other CC licenses allow remixes, but may impose conditions on how the remix may be used. For example, if you create a remix with a work licensed under a ShareAlike license, you need to make sure that the remix is licensed under the same license conditions. Likewise, if you want to use the remixed work for commercial purposes, you cannot incorporate a work released under one of the NonCommercial licenses. Review the charts below for more details on whether and how differently licensed works can be combined and licensed.
If I derive or adapt a work offered under a Creative Commons license, which CC license(s) can I apply to the resulting work?
- The chart below details the CC license(s) you may apply to the resulting work. Please note that this chart is not a substitute for legal advice and should not be relied upon as legal advice.
The green boxes indicate where compatibility exists between licenses. When modifying a work licensed under the license identified in the lefthand column, you may license the adaptation under one of the licenses indicated on the top row if the corresponding box is green. White boxes indicate those licenses that you should not use for the adapted work. In the absence of an applicable copyright exception or limitation, you cannot adapt works that contain the No Derivatives license element.
| Adapter’s license chart | Adapter’s license | |||||||
| BY | BY-NC | BY-NC-ND | BY-NC-SA | BY-ND | BY-SA | PD | ||
| Status of original work | PD | |||||||
| BY | ||||||||
| BY-NC | ||||||||
| BY-NC-ND | ||||||||
| BY-NC-SA | ||||||||
| BY-ND | ||||||||
| BY-SA |
-
Abbreviation Key
- BY = Attribution only
- BY-ND = Attribution-NoDerivatives
- BY-NC-ND = Attribution-NonCommercial- NoDerivatives
- BY-NC = Attribution-NonCommercial
- BY-NC-SA = Attribution-NonCommercial- ShareAlike
- BY-SA = Attribution-ShareAlike
- PD = Dedicated to or marked as being in the public domain via one of our public domain tools, or other public domain material; adaptations of materials in the public domain may be built upon and licensed by the creator under any license terms desired.


4 Radicals and Rational Exponents
Learning Objectives
In this section students will:
- Evaluate square roots.
- Use the product rule to simplify square roots.
- Use the quotient rule to simplify square roots.
- Add and subtract square roots.
- Rationalize denominators.
- Use rational roots.
A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in (Figure), and use the Pythagorean Theorem.
Evaluating Square Roots
When the square root of a number is squared, the result is the original number. Since[latex]\,{4}^{2}=16,[/latex]the square root of[latex]\,16\,[/latex]is[latex]\,4.\,[/latex]The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.
In general terms, if[latex]\,a\,[/latex]is a positive real number, then the square root of[latex]\,a\,[/latex]is a number that, when multiplied by itself, gives[latex]\,a.\,[/latex]The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals[latex]\,a.\,[/latex]The square root obtained using a calculator is the principal square root.
The principal square root of[latex]\,a\,[/latex]is written as[latex]\,\sqrt{a}.\,[/latex]The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.
Principal Square Root
The principal square root of[latex]\,a\,[/latex]is the nonnegative number that, when multiplied by itself, equals[latex]\,a.\,[/latex]It is written as a radical expression, with a symbol called a radical over the term called the radicand:[latex]\,\sqrt{a}.[/latex]
Does[latex]\,\sqrt{25}=±5?[/latex]
No. Although both[latex]\,{5}^{2}\,[/latex]and[latex]\,{\left(-5\right)}^{2}\,[/latex]are[latex]\,25,[/latex]the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is[latex]\,\sqrt{25}=5.[/latex]
Evaluating Square Roots
Evaluate each expression.
- [latex]\sqrt{100}[/latex]
- [latex]\sqrt{\sqrt{16}}[/latex]
- [latex]\sqrt{25+144}[/latex]
- [latex]\sqrt{49}-\sqrt{81}[/latex]
[hidden-answer a=”fs-id1441682″]
- [latex]\sqrt{100}=10\,[/latex]because[latex]\,{10}^{2}=100[/latex]
- [latex]\sqrt{\sqrt{16}}=\sqrt{4}=2\,[/latex]because[latex]\,{4}^{2}=16\,[/latex]and[latex]\,{2}^{2}=4[/latex]
- [latex]\sqrt{25+144}=\sqrt{169}=13\,[/latex]because[latex]\,{13}^{2}=169[/latex]
- [latex]\sqrt{49}-\sqrt{81}=7-9=-2\,[/latex]because[latex]\,{7}^{2}=49\,[/latex]and[latex]\,{9}^{2}=81[/latex]
[/hidden-answer]
For[latex]\,\sqrt{25+144},[/latex]can we find the square roots before adding?
No.[latex]\,\sqrt{25}+\sqrt{144}=5+12=17.\,[/latex]This is not equivalent to[latex]\,\sqrt{25+144}=13.\,[/latex]The order of operations requires us to add the terms in the radicand before finding the square root.
Try It
Evaluate each expression.
- [latex]\sqrt{225}[/latex]
- [latex]\sqrt{\sqrt{81}}[/latex]
- [latex]\sqrt{25-9}[/latex]
- [latex]\sqrt{36}+\sqrt{121}[/latex]
[hidden-answer a=”fs-id1399573″]
- [latex]15[/latex]
- [latex]3[/latex]
- [latex]4[/latex]
- [latex]17[/latex]
[/hidden-answer]
Using the Product Rule to Simplify Square Roots
To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite[latex]\,\sqrt{15}\,[/latex]as[latex]\,\sqrt{3}\cdot \sqrt{5}.\,[/latex]We can also use the product rule to express the product of multiple radical expressions as a single radical expression.
The Product Rule for Simplifying Square Roots
If[latex]\,a\,[/latex]and[latex]\,b\,[/latex]are nonnegative, the square root of the product[latex]\,ab\,[/latex]is equal to the product of the square roots of[latex]\,a\,[/latex]and[latex]\,b.\,[/latex]
How To
Given a square root radical expression, use the product rule to simplify it.
- Factor any perfect squares from the radicand.
- Write the radical expression as a product of radical expressions.
- Simplify.
Using the Product Rule to Simplify Square Roots
Simplify the radical expression.
- [latex]\sqrt{300}[/latex]
- [latex]\sqrt{162{a}^{5}{b}^{4}}[/latex]
[hidden-answer a=”fs-id1372340″]
- [latex]\begin{array}{cc}\sqrt{100\cdot 3}\hfill & \phantom{\rule{4.5em}{0ex}}\text{Factor perfect square from radicand}.\hfill \\ \sqrt{100}\cdot \sqrt{3}\hfill & \phantom{\rule{4.5em}{0ex}}\text{Write radical expression as product of radical expressions}.\hfill \\ 10\sqrt{3}\hfill & \phantom{\rule{4.5em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]
- [latex]\begin{array}{cc}\sqrt{81{a}^{4}{b}^{4}\cdot 2a}\hfill & \phantom{\rule{2em}{0ex}}\text{Factor perfect square from radicand}.\hfill \\ \sqrt{81{a}^{4}{b}^{4}}\cdot \sqrt{2a}\hfill & \phantom{\rule{2em}{0ex}}\text{Write radical expression as product of radical expressions}.\hfill \\ 9{a}^{2}{b}^{2}\sqrt{2a}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]
[/hidden-answer]
Try It
Simplify[latex]\,\sqrt{50{x}^{2}{y}^{3}z}.[/latex]
[hidden-answer a=”fs-id1308430″]
[latex]5|x||y|\sqrt{2yz}.\,[/latex]Notice the absolute value signs around x and y? That’s because their value must be positive!
[/hidden-answer]
How To
Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.
- Express the product of multiple radical expressions as a single radical expression.
- Simplify.
Using the Product Rule to Simplify the Product of Multiple Square Roots
Simplify the radical expression.
[hidden-answer a=”fs-id1301259″]
[latex]\begin{array}{cc}\sqrt{12\cdot 3}\hfill & \phantom{\rule{2em}{0ex}}\text{Express the product as a single radical expression}.\hfill \\ \sqrt{36}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}.\hfill \\ 6\hfill & \end{array}[/latex]
[/hidden-answer]
Try It
Simplify[latex]\,\sqrt{50x}\cdot \sqrt{2x}\,[/latex]assuming[latex]\,x>0.[/latex]
[hidden-answer a=”fs-id1484811″]
[latex]10|x|[/latex]
[/hidden-answer]
Using the Quotient Rule to Simplify Square Roots
Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite[latex]\,\sqrt{\frac{5}{2}}\,[/latex]as[latex]\,\frac{\sqrt{5}}{\sqrt{2}}.[/latex]
The Quotient Rule for Simplifying Square Roots
The square root of the quotient[latex]\,\frac{a}{b}\,[/latex]is equal to the quotient of the square roots of[latex]\,a\,[/latex]and[latex]\,b,[/latex]where[latex]\,b\ne 0.[/latex]
How To
Given a radical expression, use the quotient rule to simplify it.
- Write the radical expression as the quotient of two radical expressions.
- Simplify the numerator and denominator.
Using the Quotient Rule to Simplify Square Roots
Simplify the radical expression.
[latex]\sqrt{\frac{5}{36}}[/latex]
[hidden-answer a=”fs-id1392539″]
[latex]\begin{array}{cc}\frac{\sqrt{5}}{\sqrt{36}}\hfill & \phantom{\rule{2em}{0ex}}\text{Write as quotient of two radical expressions}.\hfill \\ \frac{\sqrt{5}}{6}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify denominator}.\hfill \end{array}[/latex]
[/hidden-answer]
Try It
Simplify[latex]\,\sqrt{\frac{2{x}^{2}}{9{y}^{4}}}.[/latex]
[hidden-answer a=”fs-id1424945″]
[latex]\frac{x\sqrt{2}}{3{y}^{2}}.\,[/latex]We do not need the absolute value signs for[latex]\,{y}^{2}\,[/latex]because that term will always be nonnegative.
[/hidden-answer]
Using the Quotient Rule to Simplify an Expression with Two Square Roots
Simplify the radical expression.
[latex]\frac{\sqrt{234{x}^{11}y}}{\sqrt{26{x}^{7}y}}[/latex]
[hidden-answer a=”fs-id1522001″]
[latex]\begin{array}{cc}\sqrt{\frac{234{x}^{11}y}{26{x}^{7}y}}\hfill & \phantom{\rule{2em}{0ex}}\text{Combine numerator and denominator into one radical expression}.\hfill \\ \sqrt{9{x}^{4}}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify fraction}.\hfill \\ 3{x}^{2}\text{ }\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify square root}.\hfill \end{array}[/latex]
[/hidden-answer]
Try It
Simplify[latex]\,\frac{\sqrt{9{a}^{5}{b}^{14}}}{\sqrt{3{a}^{4}{b}^{5}}}.[/latex]
[hidden-answer a=”fs-id1461508″]
[latex]{b}^{4}\sqrt{3ab}[/latex]
[/hidden-answer]
Adding and Subtracting Square Roots
We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of[latex]\,\sqrt{2}\,[/latex]and[latex]\,3\sqrt{2}\,[/latex]is[latex]\,4\sqrt{2}.\,[/latex]However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression[latex]\,\sqrt{18}\,[/latex]can be written with a[latex]\,2\,[/latex]in the radicand, as[latex]\,3\sqrt{2},[/latex]so[latex]\,\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}.[/latex]
How To
Given a radical expression requiring addition or subtraction of square roots, solve.
- Simplify each radical expression.
- Add or subtract expressions with equal radicands.
Adding Square Roots
Add[latex]\,5\sqrt{12}+2\sqrt{3}.[/latex]
[hidden-answer a=”fs-id1537392″]
We can rewrite[latex]\,5\sqrt{12}\,[/latex]as[latex]\,5\sqrt{4·3}.\,[/latex]According the product rule, this becomes[latex]\,5\sqrt{4}\sqrt{3}.\,[/latex]The square root of[latex]\,\sqrt{4}\,[/latex]is 2, so the expression becomes[latex]\,5\left(2\right)\sqrt{3},[/latex]which is[latex]\,10\sqrt{3}.\,[/latex]Now we can the terms have the same radicand so we can add.
[latex]10\sqrt{3}+2\sqrt{3}=12\sqrt{3}[/latex]
[/hidden-answer]
Try It
Add[latex]\,\sqrt{5}+6\sqrt{20}.[/latex]
[hidden-answer a=”fs-id1409688″]
[latex]13\sqrt{5}[/latex]
[/hidden-answer]
Subtracting Square Roots
Subtract[latex]\,20\sqrt{72{a}^{3}{b}^{4}c}\,-14\sqrt{8{a}^{3}{b}^{4}c}.[/latex]
[hidden-answer a=”fs-id1409634″]
Rewrite each term so they have equal radicands.
Now the terms have the same radicand so we can subtract.
[/hidden-answer]
Try It
Subtract[latex]\,3\sqrt{80x}\,-4\sqrt{45x}.[/latex]
[hidden-answer a=”fs-id1483211″]
[latex]0[/latex]
[/hidden-answer]
Rationalizing Denominators
When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.
We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.
For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is[latex]\,b\sqrt{c},[/latex]multiply by[latex]\,\frac{\sqrt{c}}{\sqrt{c}}.[/latex]
For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is[latex]\,a+b\sqrt{c},[/latex]then the conjugate is[latex]\,a-b\sqrt{c}.[/latex]
How To
Given an expression with a single square root radical term in the denominator, rationalize the denominator.
- Multiply the numerator and denominator by the radical in the denominator.
- Simplify.
Rationalizing a Denominator Containing a Single Term
Write[latex]\,\frac{2\sqrt{3}}{3\sqrt{10}}\,[/latex]in simplest form.
[hidden-answer a=”fs-id1507291″]
The radical in the denominator is[latex]\,\sqrt{10}.\,[/latex]So multiply the fraction by[latex]\,\frac{\sqrt{10}}{\sqrt{10}}.\,[/latex]Then simplify.
Try It
Write[latex]\,\frac{12\sqrt{3}}{\sqrt{2}}\,[/latex]in simplest form.
[hidden-answer a=”fs-id1167339431905″]
[latex]6\sqrt{6}[/latex]
[/hidden-answer]
How To
Given an expression with a radical term and a constant in the denominator, rationalize the denominator.
- Find the conjugate of the denominator.
- Multiply the numerator and denominator by the conjugate.
- Use the distributive property.
- Simplify.
Rationalizing a Denominator Containing Two Terms
Write[latex]\,\frac{4}{1+\sqrt{5}}\,[/latex]in simplest form.
[hidden-answer a=”fs-id1466821″]
Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of[latex]\,1+\sqrt{5}\,[/latex]is[latex]\,1-\sqrt{5}.\,[/latex]Then multiply the fraction by[latex]\,\frac{1-\sqrt{5}}{1-\sqrt{5}}.[/latex]
Try It
Write[latex]\,\frac{7}{2+\sqrt{3}}\,[/latex]in simplest form.
[hidden-answer a=”fs-id1167339432016″]
[latex]14-7\sqrt{3}[/latex]
[/hidden-answer]
Using Rational Roots
Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.
Understanding nth Roots
Suppose we know that[latex]\,{a}^{3}=8.\,[/latex]We want to find what number raised to the 3rd power is equal to 8. Since[latex]\,{2}^{3}=8,[/latex]we say that 2 is the cube root of 8.
The nth root of[latex]\,a\,[/latex]is a number that, when raised to the nth power, gives[latex]\,a.\,[/latex]For example,[latex]\,-3\,[/latex]is the 5th root of[latex]\,-243\,[/latex]because[latex]\,{\left(-3\right)}^{5}=-243.\,[/latex]If[latex]\,a\,[/latex]is a real number with at least one nth root, then the principal nth root of[latex]\,a\,[/latex]is the number with the same sign as[latex]\,a\,[/latex]that, when raised to the nth power, equals[latex]\,a.[/latex]
The principal nth root of[latex]\,a\,[/latex]is written as[latex]\,\sqrt[n]{a},[/latex]where[latex]\,n\,[/latex]is a positive integer greater than or equal to 2. In the radical expression,[latex]\,n\,[/latex]is called the index of the radical.
Principal nth Root
If[latex]\,a\,[/latex]is a real number with at least one nth root, then the principal nth root of[latex]\,a,[/latex]written as[latex]\,\sqrt[n]{a},[/latex]is the number with the same sign as[latex]\,a\,[/latex]that, when raised to the nth power, equals[latex]\,a.\,[/latex]The index of the radical is[latex]\,n.[/latex]
Simplifying nth Roots
Simplify each of the following:
- [latex]\sqrt[5]{-32}[/latex]
- [latex]\sqrt[4]{4}\cdot \sqrt[4]{1,024}[/latex]
- [latex]-\sqrt[3]{\frac{8{x}^{6}}{125}}[/latex]
- [latex]8\sqrt[4]{3}-\sqrt[4]{48}[/latex]
[hidden-answer a=”983350″]
- [latex]\sqrt[5]{-32}=-2\,[/latex]because[latex]\,{\left(-2\right)}^{5}=-32[/latex]
- First, express the product as a single radical expression.[latex]\,\sqrt[4]{4,096}=8\,[/latex]because[latex]\,{8}^{4}=4,096[/latex]
- [latex]\begin{array}{cc}\frac{-\sqrt[3]{8{x}^{6}}}{\sqrt[3]{125}}\hfill & \phantom{\rule{3em}{0ex}}\text{Write as quotient of two radical expressions}.\hfill \\ \frac{-2{x}^{2}}{5}\hfill & \phantom{\rule{3em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]
- [latex]\begin{array}{cc}8\sqrt[4]{3}-2\sqrt[4]{3}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify to get equal radicands}.\hfill \\ 6\sqrt[4]{3} \hfill & \phantom{\rule{2em}{0ex}}\text{Add}.\hfill \end{array}[/latex]
[/hidden-answer]
Try It
Simplify.
- [latex]\sqrt[3]{-216}[/latex]
- [latex]\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}[/latex]
- [latex]6\sqrt[3]{9,000}+7\sqrt[3]{576}[/latex]
[hidden-answer a=”fs-id1438231″]
- [latex]-6[/latex]
- [latex]6[/latex]
- [latex]88\sqrt[3]{9}[/latex]
[/hidden-answer]
Using Rational Exponents
Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index[latex]\,n\,[/latex]is even, then[latex]\,a\,[/latex]cannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.
Rational Exponents
Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is
How To
Given an expression with a rational exponent, write the expression as a radical.
- Determine the power by looking at the numerator of the exponent.
- Determine the root by looking at the denominator of the exponent.
- Using the base as the radicand, raise the radicand to the power and use the root as the index.
Writing Rational Exponents as Radicals
Write[latex]\,{343}^{\frac{2}{3}}\,[/latex]as a radical. Simplify.
[hidden-answer a=”fs-id1493792″]
The 2 tells us the power and the 3 tells us the root.
[latex]{343}^{\frac{2}{3}}={\left(\sqrt[3]{343}\right)}^{2}=\sqrt[3]{{343}^{2}}[/latex]
We know that[latex]\,\sqrt[3]{343}=7\,[/latex]because[latex]\,{7}^{3}=343.\,[/latex]Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.
[latex]{343}^{\frac{2}{3}}={\left(\sqrt[3]{343}\right)}^{2}={7}^{2}=49[/latex]
[/hidden-answer]
Try It
Write[latex]\,{9}^{\frac{5}{2}}\,[/latex]as a radical. Simplify.
[hidden-answer a=”fs-id1481506″]
[latex]{\left(\sqrt{9}\right)}^{5}={3}^{5}=243[/latex]
[/hidden-answer]
Writing Radicals as Rational Exponents
Write[latex]\,\frac{4}{\sqrt[7]{{a}^{2}}}\,[/latex]using a rational exponent.
[hidden-answer a=”fs-id1522098″]The power is 2 and the root is 7, so the rational exponent will be[latex]\,\frac{2}{7}.\,[/latex]We get[latex]\,\frac{4}{{a}^{\frac{2}{7}}}.\,[/latex]Using properties of exponents, we get[latex]\,\frac{4}{\sqrt[7]{{a}^{2}}}=4{a}^{\frac{-2}{7}}.[/latex][/hidden-answer]
Try It
Write[latex]\,x\sqrt{{\left(5y\right)}^{9}}\,[/latex]using a rational exponent.
[hidden-answer a=”fs-id1520416″]
[latex]x{\left(5y\right)}^{\frac{9}{2}}[/latex]
[/hidden-answer]
Simplifying Rational Exponents
Simplify:
- [latex]5\left(2{x}^{\frac{3}{4}}\right)\left(3{x}^{\frac{1}{5}}\right)[/latex]
- [latex]{\left(\frac{16}{9}\right)}^{-\frac{1}{2}}[/latex]
[hidden-answer a=”fs-id1489299″]
- [latex]\begin{array}{cc}30{x}^{\frac{3}{4}}{x}^{\frac{1}{5}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Multiply the coefficients}.\hfill \\ 30{x}^{\frac{3}{4}+\frac{1}{5}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Use properties of exponents}.\hfill \\ 30{x}^{\frac{19}{20}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]
- [latex]\begin{array}{cc}{\left(\frac{9}{16}\right)}^{\frac{1}{2}}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Use definition of negative exponents}.\hfill \\ \sqrt{\frac{9}{16}}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Rewrite as a radical}.\hfill \\ \frac{\sqrt{9}}{\sqrt{16}}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Use the quotient rule}.\hfill \\ \frac{3}{4}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Simplify}.\hfill \end{array}[/latex]
[/hidden-answer]
Try It
Simplify[latex]\,\left[{\left(8x\right)}^{\frac{1}{3}}\right]\left(14{x}^{\frac{6}{5}}\right).[/latex]
[hidden-answer a=”fs-id1517645″]
[latex]28{x}^{\frac{23}{15}}[/latex]
[/hidden-answer]
Access these online resources for additional instruction and practice with radicals and rational exponents.
Key Concepts
- The principal square root of a number[latex]\,a\,[/latex]is the nonnegative number that when multiplied by itself equals[latex]\,a.\,[/latex]See (Figure).
- If[latex]\,a\,[/latex]and[latex]\,b\,[/latex]are nonnegative, the square root of the product[latex]\,ab\,[/latex]is equal to the product of the square roots of[latex]\,a\,[/latex]and[latex]\,b\,[/latex]See (Figure) and (Figure).
- If[latex]\,a\,[/latex]and[latex]\,b\,[/latex]are nonnegative, the square root of the quotient[latex]\,\frac{a}{b}\,[/latex]is equal to the quotient of the square roots of[latex]\,a\,[/latex]and[latex]\,b\,[/latex]See (Figure) and (Figure).
- We can add and subtract radical expressions if they have the same radicand and the same index. See (Figure) and (Figure).
- Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See (Figure) and (Figure).
- The principal nth root of[latex]\,a\,[/latex]is the number with the same sign as[latex]\,a\,[/latex]that when raised to the nth power equals[latex]\,a.\,[/latex]These roots have the same properties as square roots. See (Figure).
- Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See (Figure) and (Figure).
- The properties of exponents apply to rational exponents. See (Figure).
Section Exercises
Verbal
What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.
[hidden-answer a=”fs-id1538355″]
When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.
[/hidden-answer]
Where would radicals come in the order of operations? Explain why.
Every number will have two square roots. What is the principal square root?
[hidden-answer a=”fs-id1538371″]
The principal square root is the nonnegative root of the number.
[/hidden-answer]
Can a radical with a negative radicand have a real square root? Why or why not?
Numeric
For the following exercises, simplify each expression.
[latex]\sqrt{256}[/latex]
[hidden-answer a=”fs-id1514832″]
16
[/hidden-answer]
[latex]\sqrt{\sqrt{256}}[/latex]
[latex]\sqrt{4\left(9+16\right)}[/latex]
[hidden-answer a=”fs-id1549258″]
10
[/hidden-answer]
[latex]\sqrt{289}-\sqrt{121}[/latex]
[latex]\sqrt{196}[/latex]
[hidden-answer a=”fs-id1430081″]
14
[/hidden-answer]
[latex]\sqrt{1}[/latex]
[latex]\sqrt{98}[/latex]
[hidden-answer a=”fs-id1487474″]
[latex]7\sqrt{2}[/latex]
[/hidden-answer]
[latex]\sqrt{\frac{27}{64}}[/latex]
[latex]\sqrt{\frac{81}{5}}[/latex]
[hidden-answer a=”fs-id1511192″]
[latex]\frac{9\sqrt{5}}{5}[/latex]
[/hidden-answer]
[latex]\sqrt{800}[/latex]
[latex]\sqrt{169}+\sqrt{144}[/latex]
[hidden-answer a=”fs-id1167339435244″]
25
[/hidden-answer]
[latex]\sqrt{\frac{8}{50}}[/latex]
[latex]\frac{18}{\sqrt{162}}[/latex]
[hidden-answer a=”fs-id1167339435311″]
[latex]\sqrt{2}[/latex]
[/hidden-answer]
[latex]\sqrt{192}[/latex]
[latex]14\sqrt{6}-6\sqrt{24}[/latex]
[hidden-answer a=”fs-id1167339435403″]
[latex]2\sqrt{6}[/latex]
[/hidden-answer]
[latex]15\sqrt{5}+7\sqrt{45}[/latex]
[latex]\sqrt{150}[/latex]
[hidden-answer a=”fs-id1167339435474″]
[latex]5\sqrt{6}[/latex]
[/hidden-answer]
[latex]\sqrt{\frac{96}{100}}[/latex]
[latex]\left(\sqrt{42}\right)\left(\sqrt{30}\right)[/latex]
[hidden-answer a=”fs-id1167339435576″]
[latex]6\sqrt{35}[/latex]
[/hidden-answer]
[latex]12\sqrt{3}-4\sqrt{75}[/latex]
[latex]\sqrt{\frac{4}{225}}[/latex]
[hidden-answer a=”fs-id1167339435660″]
[latex]\frac{2}{15}[/latex]
[/hidden-answer]
[latex]\sqrt{\frac{405}{324}}[/latex]
[latex]\sqrt{\frac{360}{361}}[/latex]
[hidden-answer a=”fs-id1167339435748″]
[latex]\frac{6\sqrt{10}}{19}[/latex]
[/hidden-answer]
[latex]\frac{5}{1+\sqrt{3}}[/latex]
[latex]\frac{8}{1-\sqrt{17}}[/latex]
[hidden-answer a=”fs-id1167339435848″]
[latex]-\frac{1+\sqrt{17}}{2}[/latex]
[/hidden-answer]
[latex]\sqrt[4]{16}[/latex]
[latex]\sqrt[3]{128}+3\sqrt[3]{2}[/latex]
[hidden-answer a=”fs-id1167339435944″]
[latex]7\sqrt[3]{2}[/latex]
[/hidden-answer]
[latex]\sqrt[5]{\frac{-32}{243}}[/latex]
[latex]\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}[/latex]
[hidden-answer a=”fs-id1167339436048″]
[latex]15\sqrt{5}[/latex]
[/hidden-answer]
[latex]3\sqrt[3]{-432}+\sqrt[3]{16}[/latex]
Algebraic
For the following exercises, simplify each expression.
[latex]\sqrt{400{x}^{4}}[/latex]
[hidden-answer a=”fs-id1167339436147″]
[latex]20{x}^{2}[/latex]
[/hidden-answer]
[latex]\sqrt{4{y}^{2}}[/latex]
[latex]\sqrt{49p}[/latex]
[hidden-answer a=”fs-id1167339436220″]
[latex]7\sqrt{p}[/latex]
[/hidden-answer]
[latex]{\left(144{p}^{2}{q}^{6}\right)}^{\frac{1}{2}}[/latex]
[latex]{m}^{\frac{5}{2}}\sqrt{289}[/latex]
[hidden-answer a=”fs-id1167339436339″]
[latex]17{m}^{2}\sqrt{m}[/latex]
[/hidden-answer]
[latex]9\sqrt{3{m}^{2}}+\sqrt{27}[/latex]
[latex]3\sqrt{a{b}^{2}}-b\sqrt{a}[/latex]
[hidden-answer a=”fs-id1167339436451″]
[latex]2b\sqrt{a}[/latex]
[/hidden-answer]
[latex]\frac{4\sqrt{2n}}{\sqrt{16{n}^{4}}}[/latex]
[latex]\sqrt{\frac{225{x}^{3}}{49x}}[/latex]
[hidden-answer a=”fs-id1167339436568″]
[latex]\frac{15x}{7}[/latex]
[/hidden-answer]
[latex]3\sqrt{44z}+\sqrt{99z}[/latex]
[latex]\sqrt{50{y}^{8}}[/latex]
[hidden-answer a=”fs-id1167339436659″]
[latex]5{y}^{4}\sqrt{2}[/latex]
[/hidden-answer]
[latex]\sqrt{490b{c}^{2}}[/latex]
[latex]\sqrt{\frac{32}{14d}}[/latex]
[hidden-answer a=”fs-id1167339436753″]
[latex]\frac{4\sqrt{7d}}{7d}[/latex]
[/hidden-answer]
[latex]{q}^{\frac{3}{2}}\sqrt{63p}[/latex]
[latex]\frac{\sqrt{8}}{1-\sqrt{3x}}[/latex]
[hidden-answer a=”fs-id1167339436876″]
[latex]\frac{2\sqrt{2}+2\sqrt{6x}}{1-3x}[/latex]
[/hidden-answer]
[latex]\sqrt{\frac{20}{121{d}^{4}}}[/latex]
[latex]{w}^{\frac{3}{2}}\sqrt{32}-{w}^{\frac{3}{2}}\sqrt{50}[/latex]
[hidden-answer a=”fs-id1167339437041″]
[latex]-w\sqrt{2w}[/latex]
[/hidden-answer]
[latex]\sqrt{108{x}^{4}}+\sqrt{27{x}^{4}}[/latex]
[latex]\frac{\sqrt{12x}}{2+2\sqrt{3}}[/latex]
[hidden-answer a=”fs-id1167339437161″]
[latex]\frac{3\sqrt{x}-\sqrt{3x}}{2}[/latex]
[/hidden-answer]
[latex]\sqrt{147{k}^{3}}[/latex]
[latex]\sqrt{125{n}^{10}}[/latex]
[hidden-answer a=”fs-id1167339437263″]
[latex]5{n}^{5}\sqrt{5}[/latex]
[/hidden-answer]
[latex]\sqrt{\frac{42q}{36{q}^{3}}}[/latex]
[latex]\sqrt{\frac{81m}{361{m}^{2}}}[/latex]
[hidden-answer a=”fs-id1167339437378″]
[latex]\frac{9\sqrt{m}}{19m}[/latex]
[/hidden-answer]
[latex]\sqrt{72c}-2\sqrt{2c}[/latex]
[latex]\sqrt{\frac{144}{324{d}^{2}}}[/latex]
[hidden-answer a=”fs-id1167339437491″]
[latex]\frac{2}{3d}[/latex]
[/hidden-answer]
[latex]\sqrt[3]{24{x}^{6}}+\sqrt[3]{81{x}^{6}}[/latex]
[latex]\sqrt[4]{\frac{162{x}^{6}}{16{x}^{4}}}[/latex]
[hidden-answer a=”fs-id1167339437625″]
[latex]\frac{3\sqrt[4]{2{x}^{2}}}{2}[/latex]
[/hidden-answer]
[latex]\sqrt[3]{64y}[/latex]
[latex]\sqrt[3]{128{z}^{3}}-\sqrt[3]{-16{z}^{3}}[/latex]
[hidden-answer a=”fs-id1167339437750″]
[latex]6z\sqrt[3]{2}[/latex]
[/hidden-answer]
[latex]\sqrt[5]{1,024{c}^{10}}[/latex]
Real-World Applications
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating[latex]\,\sqrt{90,000+160,000}.\,[/latex]What is the length of the guy wire?
[hidden-answer a=”fs-id1167339437865″]
500 feet
[/hidden-answer]
A car accelerates at a rate of[latex]\,6-\frac{\sqrt{4}}{\sqrt{t}}{\text{ m/s}}^{2}\,[/latex]where t is the time in seconds after the car moves from rest. Simplify the expression.
Extensions
For the following exercises, simplify each expression.
[latex]\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}}-{2}^{\frac{1}{2}}[/latex]
[hidden-answer a=”fs-id1167339438011″]
[latex]\frac{-5\sqrt{2}-6}{7}[/latex]
[/hidden-answer]
[latex]\frac{{4}^{\frac{3}{2}}-{16}^{\frac{3}{2}}}{{8}^{\frac{1}{3}}}[/latex]
[latex]\frac{\sqrt{m{n}^{3}}}{{a}^{2}\sqrt{{c}^{-3}}}\cdot \frac{{a}^{-7}{n}^{-2}}{\sqrt{{m}^{2}{c}^{4}}}[/latex]
[hidden-answer a=”fs-id1167339438268″]
[latex]\frac{\sqrt{mnc}}{{a}^{9}cmn}[/latex]
[/hidden-answer]
[latex]\frac{a}{a-\sqrt{c}}[/latex]
[latex]\frac{x\sqrt{64y}+4\sqrt{y}}{\sqrt{128y}}[/latex]
[hidden-answer a=”fs-id1167339438405″]
[latex]\frac{2\sqrt{2}x+\sqrt{2}}{4}[/latex]
[/hidden-answer]
[latex]\left(\frac{\sqrt{250{x}^{2}}}{\sqrt{100{b}^{3}}}\right)\left(\frac{7\sqrt{b}}{\sqrt{125x}}\right)[/latex]
[latex]\sqrt{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}[/latex]
[hidden-answer a=”fs-id1167339438628″]
[latex]\frac{\sqrt{3}}{3}[/latex]
[/hidden-answer]
Glossary
- index
- the number above the radical sign indicating the nth root
- principal nth root
- the number with the same sign as[latex]\,a\,[/latex]that when raised to the nth power equals[latex]\,a[/latex]
- principal square root
- the nonnegative square root of a number[latex]\,a\,[/latex]that, when multiplied by itself, equals[latex]\,a[/latex]
- radical
- the symbol used to indicate a root
- radical expression
- an expression containing a radical symbol
- radicand
- the number under the radical symbol