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GMAT Algebra Tip: Finding Square Roots Quickly

I recently had a student submit the following question:

Question: I just came across a problem where I was asked to find the side of a right triangle, x, where the base is 7 and the hypotenuse is 25. Using the Pythagorean Theorem, I solved to find that x = √576. I didn’t immediately recognize that as a square root I knew. On test day, is there a shortcut to figure out that the square root of 576 is 24?

Answer: This is a great question, and one that raises a couple strategies that will serve you well not only on this particular question, but on several other types of GMAT quantitative questions as well. Let’s take a closer look.

Strategy #1: Use what you do know to eliminate clearly wrong answers and improve your guessing odds, if not actually zeroing in on the exact right answer.

Here’s what I mean. You may not immediately recognize that the square root of 576 is 24, but if you remember the video lesson on GMAT Algebra – Pt. 2, I list the common GMAT roots and GMAT exponents that you should know to make your life a whole lot easier (and to save a lot of time!) on test day. That list includes:

  • 20² = 400 (or, 400 = 20)
  • 25² = 625 (or, 625 = 25)

If you were forced to guess, then, you would know that the square root of 576 would lie somewhere between 20 and 25, probably closer to 25. Does that make sense? Thus, If the answer choices were,

A) 22
B) 23
C) 24
D) 25
E) 26

then you could straight away eliminate answer choices A, D, and E as clearly incorrect. If forced to guess, you’d probably err on the side of answer choice C, 24, since 576 is so much closer to 625 than it is to 400.

Strategy #2: Work Backwards from the answer choices.

Rather than trying to do something difficult like find the square root of a big number without a calculator, why not just start with the answer choices, square them, and see which one equals 576?

Isn’t that so much easier?

In fact, I’ll make it even easier for you. Rather than taking the time to multiply out 22*22, or 23*23, or 24*24, just square the units digit of each answer choice to determine whether or not it even has a chance of equaling the product you’re looking for!

In other words, let’s assume that you’re wanting to test whether answer choice A could possibly be correct. Rather than squaring 22, just square the units digit (2). 2² = 4. What that tells you is that the units digit of 22 squared will also equal 4. Yet, the product we’re trying to create is 576, with a units digit of 6. Therefore, there’s no way when I multiply it out that 22 squares is going to equal 576, because the units digits don’t match up!

That should make your life a whole lot easier. In fact, you can quickly scan the answer choices at this point and see which one, when you square the units digit, will produce a units digit of 6. Answer choice C should quickly jump off the page at you since 4² = 16, with a units digit of 6! That, then, is the shortcut to figure out that the square root of 576 is 24.

Summary

Whichever strategy appeals to you the most, your takeaway message should be that on difficult or challenging situations on the GMAT, when you feel stuck, try to out-smart the GMAT. Doing the actual math is almost always the most time-consuming method. Instead, see if you can employ a non-standard technique like working backwards from the answer choices, or at worst, using common sense to eliminate a few clearly wrong answer choices to dramatically improve your guessing odds. If you learn to think like a Dominate the GMAT student rather than like a GMAT test maker, it’ll be worth a lot of extra right answers for you on test day!