Square Roots

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.

Squaring 2-Digit Numbers Ending in 5

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 multiply it by 4 (which is one higher than 3), and you get 12. Take a 25 on the end.

So, 35 squared (35 x 35) is equal to 12(front) combine with the 25(end), giving the answer 1225. 

Let's try a higher number, like 115 squared. 11 times 12 = 132. Take 25 on the end, gives us 13225!

Square Roots

Now, you can start to learn how to square root a number faster in mind. First, you should 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! 

So how do we determine the square root?

The process starts out 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. 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? We know that 85 squared is 7225. 6889 is quite obviously less than 7225, so the square root must be 83.

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. The other half of the number, 56, 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. For example, 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!

Reference's link: http://gmmentalgym.blogspot.my/2010/10/root-extractions.html#rtsqr