Estimating Square Roots

October 18, 2010

Plenty of GMAT Quantitative questions involve exponents and roots. As usual, you don't have a calculator, so you're stuck solving the problem on paper.

When square roots are involved, this can be messy. After all, do you know the square root of 39? How would you find it?

The answer is a familiar one. On the GMAT, if it's going to take a long time to find an exact answer, you don't need the exact answer! An approximation is good enough.

Memorize Squares

For all sorts of reasons, you should memorize perfect squares at least up to 12^2 = 144. It helps to know additional round numbers, such as 15^2 = 225 and 25^2 = 625.

Once squares are burned into your memory, estimating square roots is easy. Let's go back to the square root of 39. The nearest perfect squares are 36 and 49, which are squares of 6 and 7, respectively. Thus, the square root of 39 is somewhere between 6 and 7. Since 39 is closer to 36 than to 49, the square root of 39 is closer to 6 than 7.

Sure enough, the square root of 39 is about 6.24. You will never see a problem on the GMAT where you need to know that the square root of 39 is 6.24. But you may well need to know that the square root of 39 is a little bigger than 6.

More to Memorize

For large numbers, that technique is sufficient. For smaller numbers, it helps to memorize a few more facts. The square root of 2 is about 1.4, and the square root of 3 is about 1.7.

If you ever need to estimate the square root of a non-integer, convert it to a fraction. The square root of 1.5 would be a bear to calculate, but if you treat is as 3/2, you can take the square root of the numerator and denominator, leaving you with root 3 divided by root 2, or 1.7 divided by 1.4. Still not terribly friendly, but something you can estimate.

Third Roots

Occasionally, third roots come up on GMAT problems, as well. They are less common, but you can estimate values of third roots the same way.

Instead of using perfect squares, use perfect cubes. For instance, 2^3 = 8, 3^3 = 27, and 4^3 = 64. Thus, if you need the third root of 50, consider the nearest cubes. 50 is between 27 and 64, a little closer to 64. Thus, the third root is between 3 and 4, a little closer to 4, perhaps 3.6.

Once you've memorized the basics, these approximations should be almost instantaneous. Don't think like a calculator: Think like the testmaker.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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