Square Roots and the GMAT

Stacey Koprince —  October 15, 2012 — 15 Comments

gmat square rootHave 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 has two square roots, one positive and one negative, but √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, √9   denotes 3. [However] The two square roots of 9 are √9  = 3 and -√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 √9, then the answer is 3, but not -3. That leads us to our first rule.

Rule #1: √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: √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: √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. √9 does equal 3. What does √-9 equal? Nothing! We’re not allowed to have negative signs underneath square root signs, so √-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: -√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 x2 = 9). What should I do? That brings us to our next rule.

Rule #2: x2 = 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: √(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 √(x)2 = 3 to get x2 = 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, √x2 = 3 to see whether they make the equation true. If we plug 3 into the equation √x2 = 3, we get: √(3)2 = 3. Is this true? Yes: √(3)2 = √9 and that does indeed equal 3.

Now, try plugging -3 into the equation: √(-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 √9. No more negative number under the exponent! Finishing off the problem, we get √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 √(-3)2 = 3) or an invalid situation (such as √-9, which doesn’t equal any real 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.

Stacey Koprince

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Stacey Koprince is an Instructor and Trainer as well as the Director of Online Community for Manhattan Prep. She's also a management consultant who specializes in corporate strategy. She has been teaching various standardized tests for more than fifteen years and her entire teaching philosophy can be summed up in five words: teaching students how to think.

15 responses to Square Roots and the GMAT

  1. This was very helpful !!

  2. Stacey – this is very helpful. Could you please do a write up on percentages. Percentages largely appear both on Quants and on Verbal also. It would be really helpful to know how to logically think about them.

    Thanks in anticipation.

    Cheers

    • If you can give me a more specific topic area, I’d be happy to! I can’t write an article covering how percentages are used on all of quant as well as verbal because that’d be more like a book. :) (And we already have such a book – Fractions, Decimals, and Percents!) You can mention a particular GMATPrep question that you’d like me to do or tell me a *very* specific topic, one that can be covered in about 1,000 words or so. (Note, for example, that this article doesn’t cover anywhere near the full topic of square roots – just one very specific concept.)

      Also, I have covered Percentage questions in the past from GMATPrep; you may want to browse our blog our search the archive.

  3. You wrote “√-9, which doesn’t equal any rational number”
    Does √-9 equal maybe an irrational number?

    √-9 doesn’t equal any real number. Rational numbers together with the irrational numbers are called real numbers. Rational numbers are those which can be represented as a ratio between two integers. Irrational numbers are those which cannot be represented as a ratio of two integers, like √2. A number which squared gives -9 is called a complex number, and of course, out of the scope of the GMAT.

  4. Rule #3: √(x)2 = 3 means x = 3, x = -3
    In this rule, if we cancle square root with square,,,we will get base number(3)
    According to Maths-this is correct method.

    But if we slove this equation by removing first square root, then removing square, we will get two ans: +3 and -3,

    Here both methods are correct.

    So which is the correct method

    • Follow what the article says. :) The square is underneath the square root sign. If solving directly (eg √3^2), then we’d square first, then take the square root (according to the order of operations). If we’ve got a variable and are solving via algebra, then we do the opposite: √x^2 = 3, first square both sides to get rid of the square root sign: x^2 = 9, then take the square root to get rid of the square. (Also just note that the real test wouldn’t have the parentheses around the x – that’s just a formatting thing for this blog software.)

      When you talk about canceling the square and the square root, I think you’re thinking about this situation:(√x)^2. If you have a square root, you can “cancel” the sign by squaring it (and the other side of the equation). But that’s not our starting point above.

  5. Stacey

    My question is for you…where to go ahead by solving the equation or by cancling the square root with square?

    but see- we are getting two different ans for different methods, thoes I belive mathematically correct

  6. thanks a lot stacey..good explanation but still confuse for..

    (√x)^2 =√(x^2)
    Is this true???

    I mean let us see

    (x^1/2)^2= x^1 (base sign of the “x”)

    (x^2)^1/2= x^1 (base sign of the “x”)

    I get both are equal?

    • Okay, now we’re getting into math theory and not how they’d actually test us on the exam. :) Please note that everything in the original article is predicated on how the GMAT tests this subject. It’s not (really) a math lesson; it’s a GMAT lesson!

      Yes, what you wrote is mathematically correct, but you’ll never see that on the test. Just remember: if they gave you an actual number under the square root sign, take the positive root. If they’ve got variables and exponents in the mix, take both roots. That’s good enough. :)

  7. okay….good clarification :)

  8. Stacy please help to get me correct or giude me for this Manhattan Guide question

    Question number 4, Chapter number 5, manhattan Advanced GMAT Quant Guide(Strategy guide supplement)-

    The question is-
    Q4-
    What is the units digit of y?
    (1) The units digit of y2 equals 6.
    (2) The units digit of (y + 1)2 equals 5

    Answear says. statement 2 alone is sufficient, but i See if we take
    y= (-56) then (y+1)^2= (-55)^2= 3025- satisfy the condition with y’s unit digits as 6

    y= 54 then (y+1)= (55)^2= 3025, satisfy the condition with y’s unit digit as 4, so if we consider Y as negative number then we will have one more value for Y and statement 1 will not be sufficient inthis case. So I think either Y must be positive number or ans choice must be E with both statement together not sufficient to ans. the question.

    • Please just note for future that questions from any of our resources other than articles should be posted in the appropriate folder on the forums. Alternatively, if you think you’ve found a typo or error in one of the guides (as you did here, thank you!), you can email studentservices@manhattangmat.com directly. Oh, first (if you find something else you think might be an error), check our errata page (a list of already-known errors) here:
      http://www.manhattangmat.com/errata.cfm

      I agree with you that, if we use a negative number here, that changes the answer! I’ll forward this to our publishing team so that they can correct this. Thanks for the catch!

  9. Danke f¨¹r dies Schn?ppchen, das Bereich hat mich interess¨¦ wahrhaft viel. Grace hinein dies Angebote habe ich neuste Sachen entscheident gelernt, ihre ich keinesfalls kannte. Danke, bravo und Respekt.

  10. Pravin, you are wrong mate. You have to consider that in all your examples -x is an answer as well as x. GMAT does not negate mathematics. However, I will say that GMAT examiners are wrong in their position that x^2=9, for example, does not provide sufficient data to obtain an answer. It actually does: -3 and 3 are BOTH correct answers. Just as a function can evaluate to two different values for one operand ;)

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