Square Roots and the GRE
Have you ever gotten a GRE 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 GRE) deal with square roots. The Official Guide does detail these rules, but enough students have found the explanation confusing – and have complained to us about it – that we decided to write an article to clear everything up.
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 2nd edition, page 212:
“All positive numbers have two square roots, one positive and one negative.”
Hmm. Okay, so that makes it seem like we always should take two roots, not just the positive one. Later in the same paragraph, though, the book says:
“The symbol √n is used to denote the nonnegative square root of the nonnegative number n.”
Translation: when there’s a square root symbol given with an actual number underneath it – not a variable – then we should take only the positive root. This is confusing because, although they’re not talking about variables, they use the letter n in the example. In this instance, even though they use the letter n, they define n as a “nonnegative number” – that is, they have already removed the possibility that n could be negative, so n is not really a variable.
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 that 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 we take the square root of an actual number, 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.
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.
What else does the OG say?
Here’s the second source of confusion on this topic in the OG. On the same page of the book (212), right after the quotes that I gave up above, we have a table showing various rules and examples, and these rules seem to support the idea that we should always take the positive root and only the positive root. Note something very important though: the table is introduced with the text “where a > 0 and b > 0.” In other words, everything in the table is only true when we already know that the numbers are positive! In that case, of course we only want to take the positive values!
What if we don’t already know that the numbers in question are positive? That brings us to our second and third rules.
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 Quantitative Comparison problem, think of it this way: either one is a possible value for x, so both have to be considered possible values when comparing Quantity A to Quantity B.
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. Take the square root to get x = 3, x = -3 (as in our rule #2).
If you’re not sure that rule #2 (take both roots) should apply, try plugging the two numbers into the given equation, √x2 = 3, and 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 parentheses with 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 have to 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 a 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 there’s still a good chance that 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 to the GRE 2nd Edition is copyright ETS. 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 ETS.