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Negative Square Root

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by melguy » Thu Oct 27, 2016 7:35 pm
Hello

Please help with the problem below (reproduced as is : Manhattan Book 1 Chapter 6, Page 83 Q1)

If x^2 = 11 then x = Sq root 11. Indicate whether the statement is true or false.

As per my understanding we must consider both positive and negative values for the exponent but for the purpose of the exam negative square root is out of the scope of the exam. Still the answer states FALSE : if x^2 = 11 then x could be either + Sq root 11 or - Sq root 11.

Please help. Thanks
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by MartyMurray » Thu Oct 27, 2016 8:52 pm
Hi melguy.

From what I can see you provided the answer to your own question without realizing it.

Your information indicates that the x in x² = 11 could be negative or positive, while √11 is always positive.

Doesn't that answer your question?
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by Matt@VeritasPrep » Thu Oct 27, 2016 11:05 pm
You're right that the GMAT won't allow "negative square roots", but let's be sure to clarify what those are.

-√11 is a negative square root, but the GMAT is fine with this number.

√(-11) is a "negative square root", but the GMAT is NOT fine with this number.

The crucial difference comes when the negative sign is INSIDE the square root. Any √(negative) is not recognized by the GMAT, and can be ignored, but -√(positive) is fine.

So in your question, x² = 11 DOES have two solutions that the GMAT will recognize: x = √11 and x = -√11. But x² = -11 has NO solutions that the GMAT will recognize, since its roots are x = √-11 and x = -√-11.
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by Matt@VeritasPrep » Thu Oct 27, 2016 11:07 pm
It's also worth mentioning that the GMAT only recognizes positive square roots when the square root term appears. So

x² = 121

has two roots, x = 11 and x = -11.

But x = √121 only has ONE root, x = 11. This seems so random, but the GMAT will be consistent about it, so any time a √ appears, you only need to consider the nonnegative roots. (In other words, if a DS question says "x = √121", you can say that x has one and only one solution, x = 11.)
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by melguy » Fri Oct 28, 2016 3:50 am
Hi Marty

I actually thought -√11 and √(-11) meant the same thing. Hence the confusion.

CC Matt: Thanks a lot for clearing out the confusion!
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by Brent@GMATPrepNow » Sat Oct 29, 2016 6:08 am
To paraphrase Matt's great explanation, the GMAT only deals with real numbers.

The Official Guide for GMAT Review defines real numbers as follows:
All real numbers correspond to points on the number line and all points on the number line correspond to real numbers.

Another way to put it is that the GMAT does NOT deal with complex/imaginary numbers. For example, √(-1) = i (where i is a complex/imaginary number such that i² = -1). Fortunately (for most test-takers), the GMAT does not deal with these kinds of numbers.

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