Square Roots

by on October 7th, 2012

Have you ever gotten a GMAT question wrong because you thought you were supposed to take a square root and get two different numbers but the answer key said only the positive root counted? Alternatively, have you ever gotten one wrong because you took the square root and wrote down just the positive root but the answer key said that, this time, both the positive and the negative root counted? What’s going on here?

There are a couple of rules we need to keep straight in terms of how standardized tests (including the GMAT) deal with square roots. The Official Guide does detail these rules, but enough students have questioned us about the OG explanation that we decided to write an article in hopes of clearing everything up. :)

I want to mention one thing before we dive in: the vast majority of the time, both roots do count, and it’s rare to miss an official question as long as you do take both roots. You could just decide that you’re not going to worry about it and you’re going to solve normally (always taking both square roots). Many students do still stress about this topic, though, so if you’re in that group, read on!

Doesn’t the OG say that we’re only supposed to take the positive root?

Sometimes this is true – but not always. This is where the confusion arises. Here’s a quote from the OG 13th edition, page 114:

Every positive number n has two square roots, one positive and one negative, but  sqrt{n} denotes the positive number whose square is n.

Hmm. Okay, so the first half seems quite clear that, if you take the square root, you should get two values. The second half of the sentence is a bit confusing though. Read the examples they give in the next two sentences:

For example, sqrt{9}  denotes 3. [However] The two square roots of 9 are sqrt{9} = 3 and -sqrt{9} =  -3.

Huh? (The added “However” is mine, by the way.) Think of it this way: when they give us a square root symbol with an actual number underneath it – not a variable – then we should take only the positive root. If I ask you for the value of sqrt{9}, then the answer is 3, but not -3. That leads us to our first rule.

Rule #1: sqrt{9} = 3 only, not -3

If the problem gives you an actual number below an actual square root symbol, then take only the positive root.

Note that there are no variables in that rule. Let’s insert one: sqrt{9} = x. What is x? In this case, x = 3, because whenever they give us a square root symbol with an actual number underneath, we take only the positive root; the rule doesn’t change.

Okay, what if I change the problem to this: sqrt{x} = 3. Now what is x? In this case, x = 9, but not -9. How do we know? Try plugging the actual number back into the problem. sqrt{9} does equal 3. What does sqrt{-9} equal? Nothing! We’re not allowed to have negative signs underneath square root signs, so sqrt{-9} doesn’t work. The OG indicates this on page 114:

The square root of a negative number is not a real number.

Just as an aside, if the test did want us to take the negative root of some positive number under a square root sign, they’d give us this: -sqrt{9}. First, we’d take the square root of 9 to get 3 and then that negative sign would still be hanging out there. Voilà! We have -3.

I’m going to give you one more chance to bail. If you’d like, you can stop here and just remember “they give square root symbol with real number underneath, I take positive root.” That will be enough for the vast majority of applications. If you’d like to dig deeper, though, read on.

How else can this vary?

What if they don’t give us a square root symbol? Let’s say they ask for something that will require us to take the square root of 9 without showing the square root symbol themselves (perhaps they ask for x when x^2 = 9). What should I do? That brings us to our next rule.

Rule #2: x^2 = 9 means x = 3, x = -3

How are things different in this example? We no longer have a square root sign – here, we’re dealing with an exponent. If we square the number 3, we get 9. If we square the number -3, we also get 9. Therefore, both numbers are possible values for x, because both make the equation true.

Mathematically, we would say that x = 3 or x = -3. If you’re doing a Data Sufficiency problem, think of it this way: either one is a possible value for x, so both have to be considered possible values when deciding whether some piece of information is sufficient.

We’re almost done, but there’s one more (very rare) possibility. Again, feel free to skip if you’ve had enough.

Rule #3: sqrt{x^2} = 3 means x = 3, x = -3

Okay, we’re back to our square root sign, but we also have an exponent this time! Now what? Do we take only the positive root, because we have a square root sign? Or do we take both positive and negative roots, because we have an exponent?

First, solve for the value of x: square both sides of sqrt{x^2} = 3 to get x^2= 9. Now, this looks just like our rule #2: we take the square root to get x = 3, x = -3.

If you’re not sure that rule #2 (take both roots) should apply, try plugging the two answers into the original equation, Öx2 = 3 to see whether they make the equation true. If we plug 3 into the equation sqrt{x^2} = 3, we get: sqrt{3^2} = 3. Is this true? Yes: sqrt{3^2} = sqrt{9} and that does indeed equal 3.

Now, try plugging -3 into the equation:sqrt{(-3)^2}= 3. We have a negative under the square root sign, but we also have an exponent. Follow the order of operations: square the number first to get sqrt{9}. No more negative number under the exponent! Finishing off the problem, we get sqrt{9} and once again that does equal 3, so -3 is also a possible value for x. The variable x could equal 3 or -3.

How am I going to remember all that?

Notice something: the first example has either a real number or a plain variable (no exponent) under the square root sign. In both circumstances, we solve only for the positive value of the root, not the negative one.

The second and third examples both include an exponent. Our second rule doesn’t include any square root symbol at all – if we have only exponents, no roots at all, then we can have both positive and negative roots. Our third rule does have a square root symbol, but it also has an exponent. In cases like this, we’ll usually solve for both roots; if you’re not sure, check the math just as we did in the above example. First, we solve for both solutions and then we plug both back into the original equation. Any answer that “works,” or gives us a “true” equation, is a valid possible solution.

Takeaways for Square Roots:

(1) If there is an actual number shown under an actual square root sign, then take only the positive root.

(2) If, on the other hand, there are variables and exponents involved, be careful. If you have only exponents and no square root sign, then take both roots. If you have both an exponent and a square root sign, you’ll have to do the math to see, but most of the time both the positive and negative roots will be valid.

(3) If you’re not sure whether to include the negative root, try plugging it back into the original to see whether it produces a “true” answer (such as sqrt{(-3)^2} = 3) or an “invalid” situation (such as sqrt{-9}, which doesn’t equal any rational number).

* The text excerpted above from The Official Guide for GMAT Review 13th Edition® is copyright GMAC® (the Graduate Management Admissions Council). The short excerpts are quoted under fair-use statutes for scholarly or journalistic work; use of these excerpts does not imply endorsement of this article by GMAC.

3 comments

  • Nice Article Stacey. Thanks. I have a question on the explanation of Rule #3. In the explanation of sqrt((-3)^2) you have asked us to take the order of operations. Is the order of operation related to PEMDAS or is it something else? I have tried to apply some exponents rules here. First converted the square root to exponent form. That resulted in (-3)^2/2 and eventually to -3.
    Is my approach correct? Can you guide me what went wrong here?

    • The order of operations is another name for PEMDAS, yes - which means we have to deal with what's in the parentheses first. You did an exponent manipulation first.  :)

      Also, just note: the probably won't give you that particular form on the test anyway.

  • @Stacey : Thx for the note. Agreed. They will (have to) provide the -ve number within paranthesis in a scenario like this.

Ask a Question or Leave a Reply

The author Stacey Koprince gets email notifications for all questions or replies to this post.

Some HTML allowed. Keep your comments above the belt or risk having them deleted. Signup for a Gravatar to have your pictures show up by your comment.