Powers and Roots: Law of Exponents – II

We began looking at exponent rules in Part I. We’ll continue our discussion here. So enjoy the video and transcript while you absorb these important concepts.

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Let’s Continue Looking at Exponent Rules

Now we can expand the laws of exponents a little bit further. Back in the arithmetic module, we learned about the Distributive Law, and really, the Distributive Law is one of the big ones. It’s really one of the big mathematical ideas.

And of course the Distributive Law says that P to M–plus or minus N. What we can do is just multiply the P separately times each one of those terms, that is the Distributive Law.

Multiplication distributes over addition and subtraction. As it turns out, division also distributes over addition and subtraction. Much in the same way, exponents distribute over multiplication and division. If I have (ab) to the n, or (a/b) to the n, I can distribute the exponent to each factor. So (ab) to the n equals a to the n times b to the n, a divided by b, that fraction to the n = a to the n divided by b to the n.

Exponent Rules: Example

So, according to the exponent rules we can distribute an exponent across multiplication or division.

Here’s a very quick numerical example. Suppose we have 18 to the 8th. Well, we know that we could write 18 as a product. We could write it as its prime factorization. Of course, the prime factorization of 18 is 2 times 3 squared.

So 18 to the 8th = 2 times 3 to the 8th. Well, we can distribute that exponent to each one of those factors. We’ll get a 2 to the 8th and then we’ll get a 3 squared to the 8th. And for 3 squared to the 8th of course we’ll use the rule for the product of for a power to a power, which means multiply the exponents and we’ll get 2 to the 8th x 3 to the 16th.

So notice it’s very easy to go from the prime factorization of the individual number to the prime factorization of one of its powers.

Exponent Rules: Practice Problem

Here’s a practice problem for exponent rules. Pause the video and then we’ll talk about this.

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Image by triocean

Okay. In the numerator, all we’re gonna do is multiply out that 4. We’re just going to distribute it in each one of those terms. And for each one of those terms, we’re going to have a power to a power which means multiply the exponents. So we’re gonna wind up with x to the 8th, y to the 12th, then we have to deal with the division.

Well, x to the 8th divided by x to the 5th–we subtract. The exponent has to be x cubed, y to the 12th divided by y to the- 5th. That’s gonna be 12- -5, which is 12 + 5, which is 17. And that’s why we get x cubed y to the 17th. It’s important to beware of a very common and tempting trap, because it’s close to what is true.

Look Out for Traps

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Image by solar22,/sub>

Now we’re gonna talk about a trap. First of all it’s legal to distribute multiplication over addition and subtraction. That’s 100% legal. It’s legal to distribute exponents over multiplication and division. That’s 100% legal.

But it’s illegal to distribute an exponent over addition and subtraction. So that line, that’s just the distributive law. That’s 100% legal. That’s one of the fundamental patterns in mathematics. This is, it’s also a version of the distributive law. We’re distributing the exponent over multiplication and division.

exponent rules, illegal - magoosh

Image by Chad Zuber

That’s also 100% legal. The thing that is illegal, is distributing the exponent over addition or subtraction. That is always illegal. In fact, M plus or minus N to the p means that we’re taking that what’s in the parentheses M plus or minus N and multiplying it by itself p times. So these were variables, we have to foil out several times.

So you’ll never actually have to do that. But it’s just important to keep in mind that that’s what it would be, not multiplying, not raising the individual terms of these powers. And I will say this is a very tricky one because even when you understand that this third line is illegal, the human brain’s inborn pattern-matching software is tempted to make that mistake again–especially when you are under pressure.

You really need to know this cold. Then even when you walk into the test and you’re stressed in the middle of this test, you don’t accidentally fall into making this mistake again–because it is a very tempting trap.

Distributive Law with Powers

Again let’s look at all of this with numbers. Here is just the ordinary distributive law with numbers. Multiplication distributing over addition.

Here is the distributive law with powers. So that exponent distributing over multiplication and division. But it would be illegal if we had 8 plus or minus 5 to the 3rd. That would not be 8 to the 3rd plus or minus 5 to the 3rd. And one way to see this is to just think, let’s just take the subtraction case. If we look at (8- 5) to the 3rd, well what is that?

Of course that is 3 to the 3rd, which is 27. Whereas if we looked at something different, 8 cubed minus 5 cubed, well 8 cubed as we’ve mentioned in other videos is 512. 5 cubed is 125, and we subtract them, we get 387. And those two are not equal.

Distributing, or Factoring Out P

In other words, we get two different numerical answers. And that’s why we can’t set those things equal. We can do some legal math with sums or differences of power. First we need to go back to that most impressive pattern in the distributive law. Now this is very tricky, when we read this equation from left to right, we say that we are distributing P.

When we read this equation from right to left, we say that we are factoring out P. So distributing and factoring out are two sides of the same coin. It’s just a matter whether we were reading this equation going from left to right or from right to left, but it’s the same fundamental pattern. It’s also important to remember that any higher power of a base. Is divisible by any lower power of that same base.

Thus, in the sum of a higher power and a lower power of the same base, the greatest common factor of the two terms is the lower power, and this can be factored out because a lower power is always a factor of a higher power. So for example, 17 to the 30th + 17 to the 20th. Well, first of all we know that 17 to the 30th has to be divisible by 17 to the 20th.

We know one is a factor of the other. And so 17 to the 20th is the greatest common factor of these two terms. So I’m gonna factor that out, 17 to the 30th, I can write that as 17 to 20th times 17 to the 10 by the multiplication of powers law, I can write it that way. And of course 17 to the 20th, I can write that as 17 to the 20th times 1.

I factor out 17 to the 20th and I get 17 to the 20th times parentheses 17 to the 10th plus 1. And that is a factored out form of those powers. Obviously the powers here are too large to simplify any of these resultant terms, but if the two powers in the sum are closer sometimes such simplification is easy. So I will say, pause the video and see if you can simplify this.

Okay. 3 to the 32nd- 3 to the 28th. 3 to the 28th is a factor of 3 to the 32nd. In fact 3 to the 28th is the greatest common factor of these two terms. So we’re gonna express both of them as products involving 3 to the 28th.

So 3 to the 32, we can write that as 3 to the 28th times 3 to the 4th. And of course 3 to the 28th we can write that as 3 to the 28th times 1. Factor out 3 to the 28th we get 3 to the 1/4- 1. Now 3 to the 4th, that’s something we can calculate. 3 to the fourth, so one way to think about that is if you have memorized 3 to the 4th is 81.

Also 3 to the 4th is 3 squared squared. Well, 3 squared is 9, and 9 squared is 81. So that simplifies to 81. I do 81- 1, that’s 80. And so this is 80 times 3 to the 28th. We don’t often have to solve for something in the exponent, because this usually involves much more advanced ideas on exponent rules than are found on the test.

The test will expect us to know that if bases are the same we have b to the s equals b to the t, then it must mean that the exponents are equal. That will be very important in the lesson equations with exponents, which we’ll get to much later in the module.


In summary, exponents distribute over multiplication and division, and those are the patterns.

Exponents do not distribute over addition or subtraction. Those are very tempting mistake patterns. And we can simplify the sum or difference of powers by factoring out the lower power. And finally, if we have bases are equal and we have a to the m = a to the n, we can equate the exponents.

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  • Mike MᶜGarry

    Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.