Square Roots
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? Whats 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 its rare to miss an official question as long as you do take both roots. You could just decide that youre not going to worry about it and youre going to solve normally (always taking both square roots). Many students do still stress about this topic, though, so if youre in that group, read on!
Doesnt the OG say that were only supposed to take the positive root?
Sometimes this is true but not always. This is where the confusion arises. Heres a quote from the OG 13th edition, page 114:
Every positive number n has two square roots, one positive and one negative, but [pmath]sqrt{n}[/pmath]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, [pmath]sqrt{9}[/pmath]denotes 3. [However] The two square roots of 9 are [pmath]sqrt{9}[/pmath]= 3 and -[pmath]sqrt{9}[/pmath]= -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 [pmath]sqrt{9}[/pmath], then the answer is 3, but not -3. That leads us to our first rule.
Rule #1: [pmath]sqrt{9}[/pmath]= 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. Lets insert one: [pmath]sqrt{9}[/pmath]= 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 doesnt change.
Okay, what if I change the problem to this: [pmath]sqrt{x}[/pmath]= 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. [pmath]sqrt{9}[/pmath]does equal 3. What does [pmath]sqrt{-9}[/pmath]equal? Nothing! Were not allowed to have negative signs underneath square root signs, so [pmath]sqrt{-9}[/pmath]doesnt 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, theyd give us this: -[pmath]sqrt{9}[/pmath]. First, wed take the square root of 9 to get 3 and then that negative sign would still be hanging out there. Voil! We have -3.
Im going to give you one more chance to bail. If youd 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 youd like to dig deeper, though, read on.
How else can this vary?
What if they dont give us a square root symbol? Lets 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 [pmath]x^2[/pmath]= 9). What should I do? That brings us to our next rule.
Rule #2: [pmath]x^2[/pmath]= 9 means x = 3, x = -3
How are things different in this example? We no longer have a square root sign here, were 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 youre 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.
Were almost done, but theres one more (very rare) possibility. Again, feel free to skip if youve had enough.
Rule #3:[pmath]sqrt{x^2}[/pmath]= 3 means x = 3, x = -3
Okay, were 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 [pmath]sqrt{x^2}[/pmath]= 3 to get [pmath]x^2[/pmath]= 9. Now, this looks just like our rule #2: we take the square root to get x = 3, x = -3.
If youre 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 [pmath]sqrt{x^2}[/pmath]= 3, we get: [pmath]sqrt{3^2}[/pmath]= 3. Is this true? Yes: [pmath]sqrt{3^2}[/pmath]= [pmath]sqrt{9}[/pmath]and that does indeed equal 3.
Now, try plugging -3 into the equation:[pmath]sqrt{(-3)^2}[/pmath]= 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 [pmath]sqrt{9}[/pmath]. No more negative number under the exponent! Finishing off the problem, we get [pmath]sqrt{9}[/pmath]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 doesnt 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, well usually solve for both roots; if youre 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, youll have to do the math to see, but most of the time both the positive and negative roots will be valid.
(3) If youre 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 [pmath]sqrt{(-3)^2}[/pmath]= 3) or an invalid situation (such as [pmath]sqrt{-9}[/pmath], which doesnt 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.