How to Factor With Exponents

January 6, 2011

Do you know how to simplify an expression like x^7 + x^6? If so, you're ahead of the game, and you have a skill that will come in handy on many GMAT Quant questions.

If not, don't worry. Just keep reading.

You need two skills: (1) familiarity with basic exponent rules and (2) knowledge of factoring.

A fundamental exponent rule is (x^y)(x^z) = x^(y+z). In other words, when multiplying expressions with the same base, add the exponents. What many students don't know is that the rule works in reverse.

For example, x^7 = (x^3)(x^4). Or (x^2)(x^5). Or, as is relevant to the expression above: (x^6)(x^1).

Now we can move on to factoring. Since we know that x^7 is equal to (x^6)(x^1), we can rewrite the original expression as (x^6)(x^1) + x^6. Factor out the term x^6, and the result is x^6(x + 1). That's it.

In this case, it's not clear that the factored version is more useful. Here's a more practical example that focuses on the factoring aspect.

2^13 + 2^13
(2^13)(1) + (2^13)(1)
(2^13)(1 + 1) [factoring]
(2^13)(2)
(2^13)(2^1)
2^(13+1) [exponent rule]
2^14

The GMAT loves powers of 2. This example probably wouldn't be the entirety of a Quant question, but it could very possibly be a key part of one.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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