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Root Extractions

Published on Monday, October 11, 2010 in , , ,

Introduction

The idea of finding cube roots (or any roots of any power) can strike fear into any high school math student. The traditional process for extracting roots is long and arduous. The method taught here will make it so simple, it can be done in your head!

Cube Roots

Hand a calculator to someone in your audience, and ask them to put in any two-digit number (this doesn't work with numbers above 100). Either have them multiply that number times itself, and itself once again or, if it's a scientific calculator, have them hit the "Y to the Xth power" button, and then hit 3.

Have them show you the resulting total. You will now proceed to extract the root of this number.

Let's say you're given 185,193. To work from here, you'll need to know the cubes of all the numbers from 0 through 9:
13 =   1
23 =   8
33 =  27
43 =  64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729
With practice, these are easily recalled. Notice that when 0, 1, 4, 5, 6 and 9 are cubed, the end in 0, 1, 4, 5, 6 and 9 respectively! There's another pattern for 2, 3, 7 and 8. Note that 2 cubed ends in 8, and 8 cubed ends in 2. The same holds true for 7 cubed (it ends in 3), and 3 cubed (it ends in 7). This not only makes them easier to remember, but since each digit, when cubed, doesn't end in the same digit as any other number, this will make it easy to extract the root.

To find the cube root of our example number, 185,193, we start by breaking it into two smaller numbers, right at the comma: 185 & 193. Starting with the left half of the number (185), we look for the nearest single digit cube that isn't larger than it. Let's see...the closest cube from our chart is 125, the cube of 5. 5, then, must be the left half of the number. Therefore, we already know the cube root is 50-something!

Next, let's look at the right half of the number (193). This is even easier! Note that it ends in 3. Which digit, when cubed, gives a number ending in 3? 7! Since we know the leftmost digit is 5, and the rightmost digit is 7, we've got the cube root - 57! Check on your calculator, and you'll see 57 cubed is indeed 185,193!


To quiz yourself on cube roots, click here. To learn how to do fifth roots, click here.

Fifth Roots

Fifth roots are not much tougher than cube roots, thankfully. Once again, you need to know the 5th powers of the numbers from 1-9 by heart:
1^5 =      1
2^5 =     32
3^5 =    243
4^5 =  1,024
5^5 =  3,125
6^5 =  7,776
7^5 = 16,807
8^5 = 32,768
9^5 = 59,049
Look closely at the 1s digit of each number (in bold), and you'll see that each digit, 1-9, when taken to the fifth power, always ends in itself! 9 to the 5th power ends in 9, 8 to the 8th power ends in 8, and so on, all the way down to 1. This is what makes 5th roots so easy.

The first steps are similar to those of cube roots. As an example, we'll try and find the fifth root of 4,182,119,424.

First we need to break the number into groups of five digits, so we get 41,821 and 19,424. If you've properly associated each of the earlier numbers, you should quickly realize that 41,821 is larger than 32,000 (the 8th finger), but not larger than 57,000 (the 9th finger). So we know that the leftmost digit is 8 (Hmmm...80-something).

Now, for the easy part. Look at the rightmost digit of the right half of the number (the 4 in 19,424 in this case). That's the last digit! The fifth root is 84. Check on your calculator, and you'll see that 84 to the 5th power is indeed 4,182,119,424!


To quiz yourself on cube or fifth roots, click here. To start learning how to do square roots, click here.

Squaring 2-Digit Numbers Ending in 5

Before you learn how to do square roots, you'll need to learn a quick, simple trick for squaring two-digit numbers ending in a 5.

As long as you know your multiplication tables up to 10 times 10, you'll pick up on this trick instantly.

When given a number ending in 5, simply take the 10s digit, and multiply by a number one higher than itself. Take that answer, take a "25" on the end, and you've got the answer!

For example, let's say you're asked what 35 squared is. Take the 3 (the 10s digit), and multply it by 4 (which is one higher than 3), and you get 12. Tack a 25 on the end, giving you 1225. Simple, isn't it?

Let's try a higher number, like 75 squared. 7 times 8? 56. Tacking on the 25, gives us 5625!


To quiz yourself on cube roots, fifth roots or numbers ending in 5, click here. To see how to use this trick to determine square roots, click here.

Square Roots

You might think it's strange that you learned how to extract 3rd and 5th roots before square roots. However, square roots have a quality that makes them trickier than cube roots or fifth roots. You should, of course, know the squares for the numbers 1-9, but here is something you may not have noticed before:
1^2 =  1
2^2 =  4
3^2 =  9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
Notice that both 9 squared and 1 squares end in a 1. Both 2 and 8 squared end in 4, 3 and 7 squared both end in 9 and 4 and 6 squared both end in 6, as well! If we try and determine the ones digit of the square root as we did before, we're going to run into trouble.

So how do we determine the square root?

The process starts out the same way, by breaking the number up. With squares, we'll break the number up into 2-digit numbers. For example, we'll use 6889 as an example. This would break up into 68 and 89.

68 is greater than 8 squared (64), but less than 9 squared (81), so we know that the square root is somewhere in the 80s.

The other half of the number, 89, ends in 9. Here's where the tricky part comes in. The fact that the number ends in a 9 means that it could either end in a 3 or a 7. So, how do we determine whether the square root is 83 or 87?

This is where that squaring trick ends in 5 comes in handy. Regardless of the problem, you will narrow the 1s place down to one number that is greater than 5, and another that is less than 5. Here, we're asking whether the root is 83 or 87.

Obviously, if 6889 is greater than 85 squared, it can only be 87. If 6889 is less than 85 squared, it can only be 83. So which is it? Do the trick you learned on the previous page to determine that 85 squared is 7225. 6889 is quite obviously less than 7225, so the square root must be 83.

This is the same with how all square roots will proceed. Look at the thousands (if any) and hundreds digit, and determine what the nearest square root is without going over. That gives you the 10s digit of the square root. Next, narrow the ones digit down to two possibilities by looking at the 1s digit of the square. One of these numbers will be greater than 5, and the other less than 5, so use the 5-squaring trick to determine which of your two answers is correct!

As a final example, let's take 4356. 43 is greater than 6 squared, but less than 7 squared, so we know right away that the root is in the 60s. It ends in 6, so the square root is either 64 or 66. Since we can quickly determine that 65 squared is 4225, and that 4356 is greater, then the square root can only be 66.

Obviously, when a number end in 25, you know right away that the ones digit is 5, so these are even simpler. 9025? 90 is greater than 81 (9 squared), so we know the 10s digit is 9. The 25 tells us that the 1s digit is 5, so the square root of 9025 must be 95!


To quiz yourself on cube roots, fifth roots, numbers ending in 5 or square roots click here.

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5 Response to Root Extractions

February 11, 2013 at 9:18 PM

Does this method also work for 4, 5, or 7 digit numbers?

February 19, 2013 at 10:30 PM

I'm pretty sure these methods only work for perfect squares, cubes and 5ths. Have you got a way to easily extract cube roots of decimal numbers?

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January 24, 2016 at 4:48 PM

I really enjoyed reading your article. I found this as an informative and interesting post, so i think it is very useful and knowledgeable. I would like to thank you for the effort you have made in writing this article.


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July 7, 2016 at 12:14 PM

How do you do 2 or 3 digit numbers?

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