Powers and Roots: Law of Exponents – I

As we continue learning about powers and roots, we now turn to the law of exponents. This video is Part 1, so don’t forget to also watch Part 2 to complete your understanding of these concepts.

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Transcript: Law of Exponents

In this video we can start talking about some of the laws of exponents.

Now these are really big, these are really important on the test because these are patterns that are true for all numbers. The first situation concerns multiplying two powers. We have a to the n times a to the m. Now one thing that happens is, folks get stuck thinking about these things very abstractly, and they force themselves to memorize a bunch of abstract rules. And then it is hard to remember which rule is which, and they got themselves all confused.

I’m gonna urge you to think in terms of the fundamental definition of what an exponent is.

If you can go back to the fundamental definition of what an exponent is and understand the law from that perspective, then you will really understand it.

Multiplying Two Powers

So in order to understand multiplying two powers, let’s think about numbers. If I have 9 to the 5th times 9 to the 3rd, well think about what that means.

What fundamentally is 9 to the 5th? 9 to the 5th fundamentally means 9 factors of 5 multiplied together. And 9 cubed means 3 factors of 9 multiply together. So, in other words, I’m multiplying something with 5 factors of 9 times something with 3 factors of 9. And of course, when I multiply I can just put all those factors together.

And of course, what do I have there, how many factors? Well I have 5 plus 3, I have eight factors of 9. So of course, that equals 9 to the 8th. Clearly, 9 to the 5th times 9 cubed equals 9 to the 8th. Now, think about that a little more abstractly. We have the 7 to the m times 7 to the n, the first contains m factors of 7 multiplied together and this second contains n factor of 7 multiplied together.

So if I just multiply all the factors of 7 together what I’m going to get is a whole string of factors 7. And they will be m plus n factors of 7 in that string. What this means is that this should equal 7 to the m plus n. Now we can treat this entirely in variables. If I have a to the m times a to the n, think about this again. The first contains m factors of a, so I have a times a times a m times.

The second one contains n factors of a. So I have a times a times a n times. Multiply them altogether, I’m gonna get this long string of factors of a and the number of factors of a in that string will be n plus m. So in other words we can just add those two numbers. That’s the new exponent for a.

First Law of Exponents

And that right there is one of our laws of exponents. Multiplying two powers of the same base means that we can add the exponents. Now the next question concerns what happens when we divide powers. Again, don’t think in terms of just an abstract law, let’s go back and think this through in terms of the fundamental definition of an exponent. If you understand this law from the perspective of–fundamentally, what is an exponent?

Then you will really understand it. So, I’m going to suggest starting with numbers first. Let’s say 12 to the 7th divided by 12 cubed. How would we figure this out? Well, 12 to the 7th has to be seven factors of 12 multiplied together. We have to have that in the numerator.

In the denominator, we’re gonna have three factors of 12 multiplied together. So we’re gonna have something like this. Well obviously, we’re gonna get some cancellation here. We’re gonna be able to cancel, 1, 2, 3 factors of 12 in the numerator and denominator. So everything in red there, all that cancels.

And, of course, when you cancel, it just becomes 1. So it’s 1 and everything else. And so we’re just left with four factors of 12. And that’s 12 to the fourth. So 12 to the 7th divided by 12 to the 3rd equals 12 to the 4th right there. That suggests the rule.

We’ll make this a little more abstract, 12 to the m over 12 to the n. Well here we’re going to have a fraction and in the numerator of the fraction, we have m factors of 12. In the denominator, we have n factors of 12 and we’re going to assume that m is greater than n at least in this video. This means that n factors of 12 will cancel, So all the factors in the denominator will cancel, and that will remove some of the factors from the numerator.

And what will be left in the numerator after we remove those n factors that cancel, we’ll be left with m minus n factors of 12. And so what we’re gonna be left with is just 12 to the m minus n. Now we can do that entirely in variables, think through in variables we have a to the m divided by a to the n. So this is a fraction in the numerator we have m factors of a, in the denominator we have n factors.

For this video, again, we’re going to assume that m is greater than n. So all those factors in the denominator will cancel. When we take the m factors in the numerator and remove the n factors that cancel, we’re going to be left with m minus n. And that’s going to be the exponent of a. And right there is our second law of exponents.

Second Law of Exponents

When we divide powers of the same base, this means that we have to do is subtract the exponents. At this point, we can talk about a zero exponent. So far we’ve talked only about positive integers, but now we can expand to zero. What does a to the 0 mean? Using the division law we have just discussed, it wouldn’t be hard to create a zero exponent.

In other words, all we would need is to divide two powers that have the same exponent then the subtraction would lead to zero. So if I divide a cubed divided by a cubed–according to the law of exponents that we just have derived here–this means subtract the exponents which would be a to the 3 minus 3 which is a to the 0. But, of course, anything over itself must be equal one.

A to the 3 over a to the 3, that’s something over itself that has to equal 1. So, a to the 0 equals 1. Now, is this law always true? Notice, the only assumption we made in that argument was that the division was legal in the first place. Obviously, it wouldn’t be legal if a equals 0 but it would be legal for every other value.

So we can say if a is unequal to 0, then a to the 0 equals 1. And as far as what happened with 0 to the 0, don’t worry about that. That’s something we get into in more advanced areas of mathematics. You do not need to worry about that for the test. But you do need to know that for anything other than zero raising it to the zero power is equal to one.

Last Law of Exponents

The last scenario we will discuss is when we have a power to a power. So we have a to the m and that whole thing to the n. And again, we’re gonna think this through in terms of the fundamental laws of exponents. If you know this in terms of the fundamental laws of exponents you will understand it much more deeply.

So let’s start with numbers. 6 to the 5th and that whole thing cubed. Well of course 6 to the 5th, what that means fundamentally is five factors of six multiplied together. So we have five factors of six multiplied together in the parentheses and we’re cubing that.

Well what is it mean to cube something? To cube something means that we multiply it by itself three times. So we take that parenthesis and it’s that parenthesis times itself three times. So that would be expanded out what we mean with all the factors. Well how many factors do we have there? Well we have three parenthesis and each one Has five factors in it, so that means when we multiply it all together, we must have 3 times 5 or 15 factors of 6 and so of course it would be 6 to the 15th.

Now we’ll do this a little more abstractly. Inside the parenthesis we have m factors of a. When we raise this to the power of n, we have n different sets, each with m factors of a. In other words we have a total of m times n factors of a. And that means that the exponent has to be a to the m times n.

And that’s our law of exponents. Raising a power to a power results in multiplying the exponents.


So far the law of exponents we have reviewed here are: product to two powers means add the exponents, quotient of two powers means subtract the exponents, a to the 0 equals 1. And power to a power means multiply the exponents.

And once again, please do not memorize these in an abstract rote fashion. Please understand the arguments going back to the fundamental definition of what an exponent is. Understand those arguments, and think through them and then you’ll really understand why they’re true. I also want to mention some common mistakes with exponents.

Typical Mistakes to Watch Out For

Its good to know not only the patterns of what is true but also the typical mistake patterns because the test always likes to test those particular mistake patterns. First of all, notice that all these laws work if the bases of the two powers involved are the same. We cannot apply any of these rules if the bases are different.

So, for example we have things like this, we absolutely cannot apply a lot of exponent, 2 cubed times 3 to the 5th that does not equal 6 to something.

And, we have 12 of the 7th divided by 4 to the 5th that does not equal 3 to the something so there are no laws of exponents that are relevant if the bases are different. Another mistake folks make when working quickly, is to carry the operation on the powers to the exponent itself. In other words, when they’re multiplying their powers people make the mistake of multiplying the exponents, okay.

That’s a very common mistake when people are working quickly. Of course, the correct thing to do is add the exponents. Or when people are dividing powers they mistakenly divide the exponents. And again the correct thing to do is to subtract the exponents. And this is a really good one raising a power to a power because there’s all kinds of mistakes people can make.

So either they mistakenly raise the first exponent to the second exponent, so that’s one kind of mistake people can make. Another kind of mistake that people can make is they can fuse this with the product of the product of powers laws and they mistakenly add the exponents. So that’s another kind of mistake people can make. And of course what you’re supposed to do here is you’re supposed to multiply the two exponents so 61 to the 14th that would be the correct answer.

Finally, there is no law for the sum or difference of powers. So we’re adding 3 to the 4th plus 3 to the 7th or 5 to the 8th minus 5 squared. There is no fixed pattern for this. There is no single law of exponents. In a couple of lessons, we’ll learn how to use factoring out to simplify this kind of situation.

So in other words, we may see this but it’s not a simple law of exponents, it’s a somewhat more sophisticated trick that we’ll discuss in a couple videos. Overall, remember, as in any branch of mathematics, understanding means knowing not only the rule itself but also why it is true and what the common misunderstandings are. So that’s a really deep understanding if you can understand from that perspective.

In summary we talked about these laws of exponents, we talked about some common exponent mistakes, and we talked about how there is no exponent law for the sum or difference of powers.

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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.