## The Dreaded Exponent

Exponents have long been the bane of many students. And that should come as no surprise—after all, there are negative exponents, fractions as exponents, and the terrible sounding exponents of exponents. Feeling comfortable with exponents in all their various (and nefarious) guises will definitely help you test day. But, for right now, let’s stick to the fundamentals of exponents.

Meet the Base

The base is the number underneath the exponent. In , two is the base and three is the exponent. means that you are multiplying (the base) three times: .

That’s the easy part. Now, what about when we are combining bases? Take a look:

The rule is as follows: if the bases are the same and you are multiplying them, add the exponents. So . That is I keep the base the same (I don’t multiply it) and then I add the exponents.

What about ? Well, the bases are different so we cannot combine them as we did before. In this case, you would just have to multiply the long way, .

There is one exception to this different base rule: if the exponents are the same but the base is different, you can multiply the bases. In , the bases are different but both are to the fifth power. In this case, we keep the exponent the same and we multiply the bases. Therefore, .

Let’s try a few more examples:

**Adding Bases**

What about when you are adding similar bases?

Can we add the and the ? The answer is an unequivocal no. The only way you can change the above number is by factoring: . This last step is relatively advanced, and I wouldn’t worry about it too much unless you are going for a high score. Just remember, if you are adding the bases—whether the same or different—then you cannot add the exponents.

Finally, what happens when you take an exponent to an exponent?

?

In this case, you multiply the exponents while keeping the base, 4, the same.

.

Therefore, not .

Think you got it? Okay see if you can answer the next three questions correctly!

** **

**Questions:**

**Answers:**

- Cannot mix bases; leave as is

Dear Chris, does the GRE general test include exponents that are fractions? Or exponents that are square roots or cube roots? (And how do we solve those if we encounter them on the GRE general test?) Thank you!

Hi Dan,

I recommend that you read through the the ETS GRE Math Review, which provides a pretty comprehensive overview of what you are expected to know for the GRE. ETS includes an unfortunate caveat that “this review is not intended to be all-inclusive,” but in our experience they stick pretty closely to the concepts and materials in this book. There is no mention of non-integer (non-whole number) exponents here, but you should understand that when you raise x to a fraction n (e.g. x^1/2), you are really taking the nth power of x. So, x^1/2= √x and x^1/3= ∛x. I don’t think that you would see exponents that are square or cube roots that you are expected to solve, but it’s possible that you would see something like this to test your ability to simplify difficult expressions. I recommend that you stick to these basic exponent rules for now, and take note if you see any more difficult exponents in your practice!

Some of the open-ended quantitative questions on the GRE ask for a fraction answer. In those cases, are we required to put the fraction in lowest terms?

Hi Chris,

Could you please clarify the expression: 2x*x. If the base is different and the exponents are the same, you said multiply the bases and leave the exponents: so it should be 2x, but it is actually 2x^2. Why is that? Thank you!

What if the bases and exponents were equal when adding or subtracting?

3^8+ 3^8 = What does this simplify to?

Good question!

The most you can do with 3^8 + 3^8 is 2(3^8). There isn’t some nifty combining of bases and/or exponents. One good trick to know is that if you have a series of bases and exponents, such as 3^8 + 3^9 +3^10, factor out the smallest number (in this case 3^8) to simplify: 3^8(1 + 3^1 + 3^2) = 3^8(13).

Hope that helps 🙂

My question pertains to the sample question #2. I thought you can factor out a 3^7, because it has the same base of 3. The answer would be 3^7(3+1) = 3^7(4) = 8,748. Thank you so much for your time.

Hi Michelle,

Oh, you are definitely right in terms of the answer = 8,748 :). I was looking at it from a simplification standpoint. I guess you could change it to 3^7(4) is somewhat simplified (as I illustrated in one of the other questions in this post). But the GRE would never require you to multiply out the exponents to such a large number (even with the calculator nearby).