Factoring Exponential Expressions

Most of the time on the GMAT, manipulations with exponents are fairly straightforward. Usually, you’ll see two terms with a common base, and you’ll be expected to divide or multiply those terms (such as: 2⁵ / 2³ or 3⁵ x 3⁸). However, when you get to some of the higher-level questions, you’ll be expected to handle situations in which none of the exponent rules that you’ve learned will apply.

For example, you can be asked to simplify the following: 4¹⁷ + 4¹⁸

Since we’re not multiplying or dividing these terms, we know that we can’t use some of the common exponent rules. So what’s left? Factoring!

Whenever you’re adding or subtracting exponential terms that have the same base, one very helpful approach is to take out the greatest factor common to both terms. In the case above, the greatest common factor is 4¹⁷. We can therefore re-write the expression to read:

4¹⁷(1 + 4¹) =  4¹⁷(5)

In a vacuum, it might appear that this strategy wouldn’t take us too far, but it can come in very handy when you need to get the prime factorization of an expression. Take a look at the following:

If x and y are integers and 3⁷ – 3⁵ = 2^x3^y, then what is x + y?

The question is fundamentally concerned with how many times 2 and 3 appear in the prime factorization of 3⁷ – 3⁵. If we had infinite time, we could of course set out to calculate that different and then break it down to its prime factors, but such an approach is time-consuming and goes decidedly against the approach you should be developing for GMAT questions. So what should you do? How about factoring? If we factor 3⁵ out of both terms, we arrive at the expression:

3⁵(3² – 3¹), which breaks down to 3⁵(8) and ultimately 3⁵(2³). Now we can set this new product equal to the product on the right side and arrive at:

3⁵(2³) = 2^x3^y

At this point, it should be obvious that x = 3 and y = 5, giving us an answer of 8.

The real takeaway here is part-conceptual and part-strategic. On the GMAT, when confronted with situations that appear unfamiliar, your real focus should be on determining how the information in the question conforms to what you do know. It’s difficult to subtract exponential expressions, but by factoring, we can structure them in a way that enables us to deal with them as we would with any other exponent question.