This page is part of my unofficial solutions manual to the GRE Paper Practice Book (2e), a free resource available on the ETS website. They publish the questions; I explain the answers. If you haven’t worked through the Practice Book, give Section 5 a shot before reading this!

5.16: Powers of Five

When it comes to exponents, the GRE is a little like an old Dos Equis commercial: “I don’t always test exponents, but when I do, I prefer to test multiple properties in the same question.” To get started, we observe that 25 is a power of 5 and can be rewritten as 5²:

From there, we can use the product rule to combine exponents with the same base:

The final step involves a rule known as the property of equality of exponential functions, which is introduced (but not named) on p. 18 of the Math Review:

This is true for any positive base (other than 1) and for any integers a and b. Our base here is 5, and both 5x + 2 and n are integers, so this equation meets all of the criteria for applying the rule:

The answer, therefore, is (B). But take a look at answer (D) for a moment. This is the result you’d get if you accidentally used the power rule (which doesn’t apply to this problem) instead of the product rule. Sneaky, right? “Near-miss” answers like this are actually quite common in GRE algebra problems. It’s important, therefore, to be familiar with all the exponent rules given in the Math Review, and to know when to apply each of them.

Math Review Reference

For more on this topic, see the following sections of the GRE Math Review:

• 1.3: Exponents and Roots (p. 5)
• 2.2: Rules of Exponents (pp. 18-20)