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Here’s one of the most common math questions my students ask: “What’s up with negative numbers and square roots on the GMAT?” Luckily, the answer doesn’t involve a lot of complex rules. In this quick article, I’ll lay out the issues surrounding square roots and negative numbers, and share everything you need to know to handle them confidently.
If you’ve been studying for a while, or if you’ve worked your way through Foundations of Math , you probably know that there’s a strange interaction between negative numbers and exponents. If you square a negative number, the result is positive. If you square a positive number, the result is also positive. Squaring a number makes the negative sign ‘go away.’ This is where the problem with square roots comes in.
Suppose you’re looking at an equation that looks like this:
x² = 4
You want to find the value of x . Before you start writing out square root signs, think for a moment about what this equation actually means. “When you square some number, the result is positive 4. What is that number?” Well, the number could be positive 2. It could also be negative 2. There’s no way to know which. You have to give two answers: one positive answer, and one negative answer.
On the other hand, suppose you’ve got an equation that looks like this:
x =√ 4
Wait. Isn’t that just the same equation? Not quite. When the GMAT gives you a square root symbol , it’s referring to one specific value : the positive square root. In other words, on the GMAT, the square root of 4 is 2. If the GMAT gives you this equation, you only have to give one answer: the positive one.
That’s pretty weird, and it might not be what you learned in your middle school math class. But it is how square roots on the GMAT work. Here’s what you need to memorize:
If I see “ x² = something,” I think “x could be positive or negative.”
If I see the square root of a value, I think “the answer will always be positive.”
Try using those ideas to answer this Data Sufficiency question. The answer is at the end of this article.
If y = 2 4 , what is the value of x ?
(1) x² = y
(2) x = √ y
In tougher problems, you might also see variables under a square root. Because there’s a square root, you need to make sure you get the positive answer! You’ll do that by using absolute values . Here’s an example:
If x < 0, which of the following must equal -1?
All three of the answer choices contain the expression √ x² . How does this simplify? Well, it has to simplify to a positive number, because of the rules established above. It can’t just simplify to x , because the question states that x is negative! To make it positive, take the absolute value: √ x² =| x | . From here, simplify the expressions in order to find that (A) equals -1, while (B) and (C) both equal 1. The correct answer is (A).
Finally, how about that Data Sufficiency question from earlier? The right answer to that one is (B). The first statement is insufficient, because there are still two possible values for x . x could equal 4, or it could equal -4. However, the second statement includes a square root, which tells you that the answer is definitely positive. In this case, x has to be positive 4. Since there’s a single correct answer to the question, the second statement is sufficient.
From now on, don’t assume that x² = 4 and x = √ 4 mean exactly the same thing! On the GMAT, they’re close, but not a perfect match. Pay close attention to which type of equation the GMAT gives you, so you know how many solutions there should be.