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Lucas Fink

SAT Math: Combining Exponents

While the basic rules of exponents will really help on your SAT, you’re also going to need to know how to combine exponents. There are a number of common mistakes here, and they’ll all lose you points if you’re not careful.


When to add or subtract exponents



If you’re multiplying two powers with the same bases (which is x, here), then you can just add the two exponents. Let’s use actual numbers:


It’s pretty easy to see why if we expand the equation.




Similarly, if we divide powers that have a common base, then we can just subtract the exponent in the denominator from the one in the numerator.

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Be careful that you only do this when the bases are the same!


When to multiply or divide exponents



When you have a power of a power, you can multiply those exponents. Don’t add them!

Since roots are the opposite operation of powers, just like division is the opposite of multiplication, you can divide an exponent by the radical.

Again, if we use real numbers and expand it, the reasons why are pretty clear.


root{3}{2^6}=root{3}{(2*2)*(2*2)*(2*2)}= (2*2)=2^2


When to distribute exponents and roots



If you have an exponent outside of parentheses that contain two multiplied numbers, you need find the power of both factors.



(2*3)^2< data-lazy-src=

The FOIL method makes it pretty clear why that doesn’t work.

Radicals follow the same rules. If you have numbers under the radical that are added, then you can’t just find the root of each one. You have to combine them first.

But you can take a number under a radical, break it into factors, and simplify it that way.


And that comes in handy on the SAT.


Keep in mind:

Don’t treat exponents and radicals like other operations—they have their own set of rules to follow.


About Lucas Fink

Lucas is the teacher behind Magoosh TOEFL. He’s been teaching TOEFL preparation and more general English since 2009, and the SAT since 2008. Between his time at Bard College and teaching abroad, he has studied Japanese, Czech, and Korean. None of them come in handy, nowadays.

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