Common Exponents in GMAT Data Sufficiency

January 31, 2007

There aren’t very many specific GMAT questions that are worth articles of their own, but I’m going to focus on one of those today. There are plenty of variations; let’s start with one example, #26 from my Data Sufficiency: Challenge problem set:

Is x > 0 ?

(1) x < x^2

(2) x < x^3

(Note: “^” indicates an exponent, so “x^2″ means “x to the second power” or “x squared.”)

You’ve probably seen something like it. Usually, test-takers attack this type of question by trying out various numbers for x and seeing which ones work. Given the prevalence of this sort of question, you can learn a better way.

“Picking Numbers”

The toughest part about choosing values for x is knowing

a) which numbers to pick, and

b) when you’re done

If it’s possible to learn an approach that avoids picking numbers, it’s to your advantage to do so. On questions that deal with exponents and have no coefficients, you can avoid those problems. Instead, you can focus on three numbers: -1, 0, and 1.

Zones

You’re not going to try those exact numbers; you’re going to try numbers in the zones defined by those numbers. For instance, you want to think about one value of x that’s less than -1, one value that’s between -1 and 0, one value between 0 and 1, and finally, one value greater than 1.

Consider how x, x^2, and x^3 behave in those four “zones:”

  • x = -2: x^2 = 4, x^3 = -8. Summary: x^3 < x < x^2
  • x = -1/2: x^2 = 1/4, x^3 = -1/8. Summary: x < x^3 < x^2
  • x = 1/2: x^2 = 1/4, x^3 = 1/8. Summary: x^3 < x^2 < x
  • x = 2: x^2 = 4, x^3 = 8. Summary: x < x^2 < x^3

It may not be an effective use of your time to memorize each and every one of these relationships, but if you’re familiar with these “zones,” you’ll know exactly what numbers to test.

In Practice

Let’s look at the sample question one more time:

Is x > 0 ?

(1) x < x^2

(2) x < x^3

First, statement (1). x is almost always less than x^2, except when x is a positive number between 0 and 1. In other words, statement (1) says: “x is not between 0 and 1.” In that case, it’s insufficient: x could be positive or negative.

Statement (2) is a bit trickier. Referring to our summaries above, x is less than x^3 when x is between -1 and 0, and when x is greater than 1. Given that x could be in either one of those zones, (2) is also insufficient.

Taken together, the statements are still insufficient. (1) tells us that x could be anything less than 0 or larger than 1. (2) narrows that down somewhat: x could be between -1 and 0 or greater than 1. Thus, x could be positive or negative. Choice (E) is correct.

In General Use

Anytime you see variables (without coefficients) in a question like this, you can use these zones. Note that we aggressively simplified each statement: after we evaluated statement (1), we ignored what it actually said, and focused on what we discovered: that x couldn’t be between 0 and 1. Reducing complex expressions to simple statements makes questions like these much easier.

Jeff Sackmann is a GMAT tutor based in New York City. He has created many resources for GMAT preparation, including the popular GMAT Math Bible and GMAT Verbal Bible, as well as 1,800 practice GMAT math questions.

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