The Square Root of 1764 Is...

I was watching Arthur earlier today (yes I still watch that show) and Buster asked "the Brain" what was the square root of 1764. Unlike most people, I knew the answer without having to find a calculator.

The question of "What is the square root of ...?" is a common question people ask geniuses. I won't lie, I used to get this question all the time because I was a math whiz, but as a child, I was unable to recognize the simple pattern that existed in numbers. As a result, I never really knew the correct answer, just an estimate.

As I grew older, I soon realized that finding the square root of a number, which would have a rational number as the answer, was actually very simple, and here's how you can show your intellectual superiority to your friends by finding the square root of a number without a calculator, using the example of 1764.

When finding the square root of a number, there are two key components: the last digit (4) and the entire number itself (1764).

The last digit of the number can be used to determine the final digit of the square root. Since numbers are merely a set of patterns, you need only know the squares of the first 9 numbers (0 will always end in 0). The table below illustrates this point.

1 x 1 = 1	11 x 11 = 121	21 x 21 = 441
2 x 2 = 4	12 x 12 = 144	22 x 22 = 484
3 x 3 = 9	13 x 13 = 169	23 x 23 = 529
4 x 4 = 16	14 x 14 = 196	24 x 24 = 576
5 x 5 = 25	15 x 15 = 225	25 x 25 = 625
6 x 6 = 36	16 x 16 = 256	26 x 26 = 676
7 x 7 = 49	17 x 17 = 289	27 x 27 = 729
8 x 8 = 64	18 x 18 = 324	28 x 28 =784
9 x 9 = 81	19 x 19 = 361	29 x 29 = 841

Using the above method, 1764 has only two possible outcomes: _2 or _8.

The second component now lies within the enitre number itself. Since most people are familiar with the squares of the first ten digits, even twelve, we can use this information to determine that the nearest squares (of a multiple of ten) are 1600 (40x40) and 2500 (50x50). At this point, we realize that the first digit must be a 4_, since anything above 2500 would start with a 50, but because there are two numbers whose square ends in a 4, we must now determine if it is either 42 or 48.

The "distance" of the entire number will result in the final answer. In this case, the distance of 1764 is closer to 1600 than it is to 2500, indicating that the number is closer to 40, than it is 50. Therefore, the answer must be 42!

Here are some other examples for you to try:
3136, 6561, and 1849. You should get 56, 81, and 43. Try getting a friend to ask you what is the square root of ...? and see if you can solve it without using a calculator. As you can see, it's quite easy.