Mental Math: Exponents

Published on Sunday, February 23, 2014 in , , , , ,

Andy0101’s Wheat Chessboard imageThere's plenty of mental math shortcuts out there for addition, subtraction, and multiplication. The mental shortcuts for division are harder to find, but they are out there.

When you start asking about mental math shortcuts related to exponents, however, the only methods you find relate to taking roots, like my root extraction tutorial in the Mental Gym. What about actually taking numbers up to various powers instead?

WRAPPING YOUR HEAD AROUND EXPONENTS: Working out exponents seems scary, because most people are familiar with how quickly exponents become large numbers. You've probably heard a version of the classic legend about the reward given to the inventor of chess:

Too many people think of exponents as just repeated multiplying, but it's better to think of it as growth for a given amount of time. BetterExplained.com's article, Understanding Exponents (Why does 0^0 = 1?) is a great place to begin intuitively understanding the nature of exponents.

I've written before about mathematician and engineer Solomon Golomb, who challenged himself to be able to work out, or at least recall, the solution to any problem of the form xy where x and y were any integers (whole numbers) from 1 to 10. In college, a professor mentioned the number 224, joking that "everybody knows what that is." When Solomon Golomb realized that 224 was the same as 88, he was able to immediately reply, "Yes! It's 16,777,216!" The teacher was stunned to learn that it was the correct answer!

I may not get you that far in this post, but I think you'll be surprised just how far you can go with just a little extra mental training.

EASY EXPONENTS: There are a few easy exponents with which you should already be familiar. 1 to any power is always 1. Similarly, 10 to any power is simply 1 followed a number of zeroes equal to the power. 102 is 1 followed by 2 zeroes, or 100, 103 is 1 followed by 3 zeroes, or 1,000, and so son. Also, any number to the first power is always itself.

Squaring, of course, is just multiplying a number by itself. When you learned multiplication starting with times tables, that included multiplying all the numbers from 1 to 10 by themselves, so from this point on, I'll assume you know these already.

CUBES: When a number is raised to the 3rd power, we say it's cubed. A cube is easily worked out by multiplying a number times it's own square. 23 can be determined by multiplying 2 by the 4 (the square of 2) to get 8, and 33 can be determined by multiplying 3 × 9 (the square of 3) to get 27.

After that, we start having to deal with 2-digit numbers by 1-digit numbers. 43 is 4 × 16, and you might be fine working that out in your head, but what about 53 (5 × 25) through 93 (9 × 81)? Fortunately, Mental Math secrets is here to help you learn how to multiply 2-digit numbers by 1-digit numbers:

Once you've mastered this, you can easily work out cubes in your head! Even better, since we're only talking about 10 different cubes, regular practice will make them easy to recall, instead of calculate, so you can get them even quicker.

4TH POWER: Taking a number to the 4th power can be done by squaring the number, and then squaring the result you get. Once again, if you know your times tables, 2 and 3 are easy to take the 4th power. To get 24, we square 2 to get 4, and then square 4 to get 16. For 34, we square 3 to get 9 and then square 9 to get 81.

The problem comes, of course, when we need to square 2 digit number. For 44, it's easy enough to do 4 squared to get 16, but how do we square 16? This problem continues up to 94, for which you need to work out 81 squared. Fortunately, Harvey Mudd College Professor Arthur Benjamin is here to show you how to easily square 2-digit numbers:

Notice what the method of moving to the nearest multiple of 10 does - it turns 2-digit times 2-digit multiplications into 2-digit times 1-digit multiplication, which you learned how to do earlier! In fact, you can use those same techniques to help you work out these problems quickly.

Try this technique and practice working out the numbers 1 through 10 to the 4th power. You'll be amazed how quickly you're able to calculate, and before long just recall, each of these numbers.

5TH POWER: Once you're confident in your ability to work out 4th powers of the numbers in your head, you're ready for 5th powers.

To work out the 5th power of a number, you're going to take that number and multiply it by its own 4th power. 35 is 3 × 34, or 3...9...81, or 243. This is easy, as you've already mastered multiplying 2-digit numbers by 1-digit numbers by this point.

Multiplying numbers with 3 or more digits by a 1-digit number is similar to the process you learned earlier, but you do need to ready to work with more digits. Mental Math Secrets posted this 3-digit by 1-digit multiplication video on their site to help you learn this technique more effectively. Once you get the hang of this, multiplying 4-digit numbers by 1-digit numbers also shouldn't be that difficult.

5th powers also have a neat pattern: Each number 1 through 10, when taken to the 5th power, ends with the same digit as the number with which you started. 25 ends in 2, 35 ends in 3, and so on.

6TH POWER: If you've made it to the point where you can do 5th powers in your head, you may want to stop there. If you're ready for another challenge, however, then you may want to consider learning 6th powers.

To take a number to the 6th power, you'll need to find a number's cube, and then square it. Remember, however, that some of the cubes up to 9 are 3-digit numbers, so this will require the ability to square 3-digit numbers.

Squaring 3-digit numbers was briefly described above in the Arthur Benjamin video, but there are a few points you should know before you practice. First, you should be able to square all the numbers from 1 through 50 as fast as possible. Your adjustments will always be up or down by no more than 50, so this is an essential skill. Second, you'll need to be comfortable multiplying 3-digit numbers by 1-digit numbers.

Finally, when applying Professor Benjamin's technique for squaring 3-digit numbers, there's an easy to way to get that 2nd number you're going to multiply by that multiple of 100. Simply take the last two digits of the number you're squaring and double them, using the hundreds digit plus the final two digits of this doubling (if the number is greater than 100).

For example, when trying to square 729, you're obviously going to multiply by 700, but what's the other number? Simply double 29 (the last 2 digit of 729) to get 58, so you know the other number you need must be 758. What about 343? 43 doubled is 86, so you'll multiply 300 times 386. Once you see the pattern, it's easy to grasp.

Taking the numbers 1 through 10 to the 6th power is as far as I'm going in this post, so you can practice to the level you want. If you find any handy tips or tricks for doing exponents in mental math, I'd love to hear about them in the comments!

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1 Response to Mental Math: Exponents

12:44 PM

Awesome tutorial! I never thought I had the ability to do anything close to this! Math is a lot of fun with the right teachers!