Square Root of Perfect Squares

Today I’ll be talking about how to find the square root of a perfect square. First off, what is a perfect square? It’s a number who’s square root is an integer (e.g. 4×4, 5×5, etc.). For example, 4 and 9 are perfect squares since the square root of ’4′ is 2, and the square root ’9′ is 3.

Before we begin finding square roots, we first need to know at least the first ten perfect squares. If you need to, use the squaring method I showed you yesterday. Here they are:

Integer:    1       2       3       4       5       6       7       8       9       10

Squared:    1       4       9      16      25      36      49       64      81     100

Knowing the first ten perfect squares will allow you to calculate the square root of a lot of perfect squares (up to the number 10,000) as you will see. Ideally though, it would be nice to know the first 20 perfect squares…this will allow you to calculate the square root of perfect squares up to the number 40,000 (i.e. 200 x 200).

All right, a few things you should take notice of from the list of perfect squares before we begin. Out of the list below, #1 allows us to check and see if its even possible for the number to be a perfect square. You won’t immediately notice the significance of points #2 and onwards until we begin the examples, but just keep them in mind for the time being.

1) No perfect square can end in the numbers 2, 3, 7 or 8. Why? Because if you look at the numbers above, none of the squares end in those numbers.

2) If the Perfect Square you are looking at ends in the number ’1′, then the last digit of the square root will be either ’1′ or ’9′. If you look at the list of perfect squares above (for num 1-10), you’ll see that the squares of ’1′ and ’9′ end in a ’1′. (Also, keep in mind 1+9 = 10…I’ll explain later)

3) If the Perfect Square you are looking at ends in the number ’4′, then the last digit of the square root will be either ’2′ or ’8′. (See list of perfect squares above…the last digit of the squares of ’2′ and ’8′ end in a ’4′) Also note…2 + 8 = 10. You seeing a pattern?

4) If the Perfect Square ends in ’9′, then the last digit of the square root will be either ’3′ or ’7′. Yet again…3 + 7 = 10.

5) If the Perfect Square ends in ’6′, then the last digit of the square root will be either ’4′ or ’6′. Note…4 + 6 = 10.

6) If it ends in ’5′, the last digit will also be ’5′. If it ends in ’0′, the last digit will be ’0′.

So why did I have you keep in mind that the pairs equal 10 (1 and 9; 2 and 8; 3 and 7; 4 and 6)? It’s just a little something I found useful. It allows you to only have to remember the first ’4′ squared numbers since you can derive their pairs easily. For example: If the Perfect Square ends in ’4′, then there’s two possibilities the last digit could be (as #3 in the list indicates). What squared equals ’4′…the answer is 2. To find the other answer it could be just take 10 – 2 = 8. So it could be either 2 or 8. I’m not sure if you’ll find it helpful, but it helps me so I figured I would share it.

Anyways, if I haven’t lost you yet…on to some examples!

Example #1: Find the square root of 6084.
The first thing we should do before we even begin is split the number up into pairs of 2 digits starting from the right: so 6084 becomes 60 84. If for example, the number we were dealing with had an odd number of digits (e.g. 144) then we would still split the 44 off first…but then we would add a zero in front of the ’1′. This would turn 144 into.. 01 44.

Back to the example. Since our number is now 60 84, we count how many pairs we have. In this case we have two pairs of 2 digits. This means that our answer is going to be a two-digit number.

1) First thing we need to do is look at the last pair (’84′). The last digit in this case is ’4′. Using the list above, we see that in order to get the last digit to be ’4′ the number squared has to end in either a ’2′ or ’8′. So ’2′ and ’8′ are the possibilities for the last digit of our answer.

2) In order to get the first digit of our answer, we have to look at the first pair…in this case ’60′. What we want to do is find out between what pair of perfect squares this number falls. If you look at our initial list of Perfect Squares (for the numbers 1 to 10), you will see that the numbers that straddle ’60′ are…’49′ and ’64′.

3) Once you identify the two numbers that straddle the first pair of digits (’60′), you take the smaller one. So in this case we would keep 49 and discard 64.

4) Take the square root of 49…which is 7. This is the first digit of our answer.

5) Now look at the first group of digits (’60′) and the ones that straddle it (’49′ and ’64′). Which number is ’60′ closer to? In this case it’s closer to ’64′. Since we chose the larger number, this means that we choose the larger number of the two possible choices for our last digit (remember the choices were ’2′ and ’8′). So we choose ’8′.

Answer: The square root of 6084 is 78.

Let’s do another example, just to make sure you understand the steps.

Example #2: Find the square root of 2809.
1) The last digit is ’9′…so the last digit of the answer is either ’3′ or ’7′.
2) Out of the perfect squares in the list, which ones straddle ’28′? In this case, ’25′ and ’36′.
3) Choose ’25′ since its the lower one. The square root of ’25′ is ’5′. ’5′ is the first digit of our answer.
4) Which number is ’28′ closer to…’25′ or ’36′? It’s closer to ’25′ in this case, so we choose the smaller of our choices for the second digit. Our choices are ’3′ and ’7′; so we choose ’3′.

Answer: The square root of 2809 is 53.

Now an example which will have a three-digit answer. The list of perfect squares at the beginning will have to be extended to ’20′ for this portion instead of just up to ’10′. I’ll provide the numbers so you don’t have to worry about that…but like I said earlier, it would be a good idea to know up until 20.

Example #3: Find the square root of 16,129.
We split it up so that it becomes 01 61 29. Our answer will be a three-digit number. For the rest of the problem we will group all the digits together except the last two; so it becomes 161 29. The only reason we split it up into groups of two originally was to find out how many digits our answer was going to be…however, if you want you can use one of these formulas to find it out instead. n/2 or (n+1)/2….where n = number of digits of the perfect square. (use n/2 when the number of digits is even; else use the other)

1) The number ends in ’9′, so our choices for the last digit are ’3′ or ’7′.
2) What two numbers straddle ’161′? ’144′ (12 x 12) and ’169′ (13 x 13) straddle it. We pick ’144′ since its lower.
3) The square root of ’144′ is ’12′. So ’12′ is the first part of our answer.
4) Since ’161′ is closer to ’169′ than ’144′; we choose the larger of our possible choices for the second digit…in this case its ’7′.

Answer: The square root of 16,129 is 127.

Remember though, this only works for numbers that are Perfect Squares! There are methods that will find the square root of any number, but that’ll be in a different post. In Vedic Mathematics there is always the “General Case” which applies to all the possibilities. Then there are the “specific cases” which just speed things up for the special instances. This post deals with a “specific case”. Once you become more comfortable with it, and are able to notice patterns…you’ll be able to choose the method which is right for the situation a lot quicker.

As the last example showed, knowing the first ten perfect squares is only good up to the number 10,000. If you want to find the square root of perfect squares up to the number 40,000 you need to know the first 20 perfect squares. The next 10 are…121, 144, 169, 196, 225, 256, 289, 324, 361, and 400.

Practice Problems
Find the square root of:
1. 7396
2. 2116
3. 8836

And a little harder:
4. 23,716
5. 16,641
6. 37,249

If you need any help, or found any errors let me know!