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SquareRoots

Introduction

When you multiply a number times itself, the result is called a square number. The square root of a square number would be the original number that was multiplied by itself. If you multiply 5 * 5 (also written as 5^2^), you get 25, a square number. Therefore, the square root of 25 would be 5.

After reading and practicing the techniques in this article, you will be able to, given the square of any number from 1-99, be able to determine the square root in your head.

Learning the One-Digit Squares

To start off, you must learn and remember each of the one-digit squares:

X X^2^
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81

Learning the full square will be helpful in determining the square roots later on. Also, pay attention to the digit in the ones place in each number. Notice that the numbers 1, 4, 9 and 6 each appear twice in the ones category, once in a number below 5, and once in a number above 5. Both these keys are necessary for determining square roots in your head.

Determining the Tens Digit of the Root

To determine the tens digit of the root, you need to look at the thousands and hundreds place of the square number. For example, if you are given the number 1849, you would only need to look at the 18.

Once you have this number, you need to determine the highest sqaure that will fit into it without going over that number. The square root of that will be the tens digit. In our example, 16 is the highest square we have that is equal to, or lesser than, 18. Because we know that the square of 16 is 4, we know that the tens digit of the square root of 1849 is 4.

Determining the Ones Digit of the Root

To determine the ones digit of the root, we only need to look at the ones digit of the number you are given. If the ones digit is 5 or 0, then the ones digit of the square root will be 5 or 0 respectively, otherwise, there are some extra steps.

If the ones digit of the given number is a 1, 4, 9 or 6, then there are two possibilites for the ones digit, and you're going to need to narrow down the possibility of whether the ones digit of the root is greater than or less than 5.

In our 1849 example, we've already determined that the tens digit of the root is 4, but since the last digit is 9, the ones digit could be either 3 or 7.

To figure out which one is the correct digit, you must first learn a simple method for squaring two-digit numbers ending in 5.

Squaring Two-Digit Numbers Ending in 5

When given a two-digit number ending in 5, simply multiply the number in the tens digit by a number one greater than itself, and add a "25" on the end.

If you're trying to square the number 65 in your head, you would simply multiply 6 (the tens digit of the number in question) times 7 (a number one greater than 6), and get 42. Adding a 25 to the end of this number gives 4225, which is 65^2^!

To figure out what 85^2^ is, you simply multiply 8 time 9 (72), and put that 25 on the end, giving us the answer 7225.

Narrowing Down Between Two Possibilities

In our example of the number 1849, we've already determined that the answer is either 43 or 47, but aren't sure which yet. To determine this we need to figure our whether the given number is greater than or less than 45^2^.

Using the above procedure we determine that the square of 45 is 2025 (4x5=20, with 25 on the end). 1849 is less than 2025, so we now know that the root must be the lower of the two options - 43!

More Examples

Let's start with 1225. Looking at just the 12, we see that the highest number we can see that 9 is the highest square equal to or less than this number, so the tens digit is 3. Because the number ends in 5, we know that the square root of 1225 is 35.

As another example, we'll choose 3364. Starting with 33, we determine that 25 is the highest square, so the tens digit is 5. The ones digit of the given number is 4, so the ones digit of the square root is either 2 or 8. Is the square root 52 or 58? We have to determine 55^2^ to figure this out. 55^2^ is 3025, and 3364 is greater than 3025, so the number must be the higher of the two. That means 58 is the square root.

Estimating square roots

Suppose you are given a number that's greater than four digits long, or that isn't a perfect square? In such cases it is often useful to come up with an estimate of the square root, using some ideas which are similar to those used above.

As above, we need to pair up the digits of the target number, going away from the decimal point. Each pair of digits will in fact represent a digit in the square root. And, just like above, the leftmost pair of digits (or single digit) is used to compute the leftmost (most significant) digit of the result, simply by determining the highest square that will fit into it without going over. So, for instance, to compute the square root of 500,000, we would pair up the digits like:

50 00 00

The biggest square that fits in 50 is 49, which has a square root of 7. So we know our square root will have the form "7 - digit - digit", or "seven hundred and something". Further, since 50 is very close to 49, we can surmise that the square root will be in the low 700's. Had it been close to the next square, we would know that the square root would be in the high 700's.

It was recently reported that the ozone hole over Antarctica has shrunk to 6,000,000 square miles. How big is this? Breaking this up into pairs of digits gives you:

6 00 00 00

So you should be able to determine, with a moment's reflection, that this represents a square area measuring in the middle 2000s on each side. (If you'd rather picture a circle, it will always have a diameter that's about 13% bigger than the square root you just estimated.)

What about numbers smaller than one? Basically, the same method applies. You still pair up digits going away from the decimal point, and the most sigificant digit will still be the square root of the biggest square that fits into the leftmost pair of digits. So, to compute the square root of 0.0038234, you would pair up the digits like so:

00 . 00 38 23 40

The biggest square that fits in 38 is 36, with a root of 6. So your square root will be 0.06... Since 38 is close to 36, you can also surmise that the next digit will be small.

Other Resources

CubeRoots
CubingNumbers
FifthRoots