Powers of products & quotients (integer exponents) | Mathematics I | High School Math | Khan Academy

Do some example raising exponents or products of exponents to various powers, especially when we're dealing with integer exponents.

So let's say we have (3^8 \cdot 7^{3}), and we want to raise that to the (-2) power. I want you to pause this video and see if you could simplify this on your own.

The key realization here—there's a couple of ways that you can tackle it, but the key thing to realize is if you have the product of two things and then you're raising that to some type of an exponent, that is going to be the same thing as raising each of these things to that exponent and then taking the product.

So, this is going to be the same thing as (3^8) to the (-2) times (7^{3}) to the (-2). I'll do (7^{3}) right over here. If I want to simplify this (3^8) to the (-2), we have the other exponent property that if you're raising to an exponent and then raising that whole thing to another exponent, then you can just multiply the exponents.

So this is going to be (3^{-8 \cdot -2}). Well, (8 \cdot -2) is (16), so this is going to be (3^{16}) right over there. Then this part right over here, (7^{3}) to the (-2), that's going to be (7^{3 \cdot -2}), which is (7^{-6}) power.

So that is (7^{-6}), and this would be about as much as you could simplify it. You could rewrite it different ways: (7^{-6}) is the same thing as (\frac{1}{7^{6}}). So you could write it like (\frac{3^{16}}{7^{6}}). But these two are equivalent, and there are other ways that you could have tackled this.

You could have said that this original thing right over here, this is the same thing as (3^{8}) is the same thing as (\frac{1}{3^{8}}), so you could have said that this is the same thing as (\frac{7^{3}}{3^{8}}) and then you're raising that to the (-2), in which case you would raise this numerator to the (-2) and the denominator to the (-2), but you would have gotten to the exact same place.

Let's do another one of these. So let's say, let me, so let's say that we have (a^{-2} \cdot 8^{7}) and we want to raise all of that to the second power. Well, like before, I can raise each of these things to the second power, so this is the same thing as (a^{-2}) to the (2) power times this thing to the second power (8^{7}) to the (2) power.

Then here, (-2 \cdot 2) is (-4), so that's (a^{-4} \cdot 8^{7 \cdot 2}) which is (8^{14}). In other videos, we go into more depth about why this should hopefully make intuitive sense. Here you have (8^{7} \cdot 8^{7}), well you would then add the two exponents, and you would get to (8^{14}).

So, however many times you have (8^{7}), you would just keep adding the exponents or you would multiply by seven that many times. Hopefully, that didn't sound too confusing, but the general idea is if you raise something to an exponent and then another exponent, you can multiply those exponents.

Let's do one more example where we are dealing with quotients, which that first example could have even been perceived as. So let’s say we have (2^{-10} \div 4^{2}) and we're going to raise all of that to the seventh power.

Well, this is equivalent to (\frac{2^{-10}}{4^{2}}) raised to the seventh power. So if you have the difference of two things and you're raising it to some power, that's the same thing as the numerator raised to that power divided by the denominator raised to that power.

Well, what's our numerator going to be? Well, we've done this drill before: it would be (2^{-10 \cdot 7}) so this would be equal to (2^{-70}). And then in the denominator (4^{2}) raised to the seventh power.

Well, (2 \cdot 7) is (14), so that's going to be (4^{14}) power. Now we actually could think about simplifying this even more. There are multiple ways that you could rewrite this, but one thing you could do is say, “Hey look, two (4)s are a power of two.”

So you could rewrite this as (2^{-70}) over instead of writing (4^{14}) power. Why did I write (14)th power? It should be (4^{14}) power. Let me correct that: instead of writing (4^{14}) power, I instead could write...

So, this is (2^{-70}) over instead of writing (4) I could write (2^{2}) to the (14)th power. (4) is the same thing as (2^{2}). And so now I can rewrite this whole thing as (2^{-70}) over, well, (2^{2}) then that to the (14)th power that's (2^{28}) power.

So can I simplify this even more? Well, this is going to be equal to... If I'm taking a quotient with the same base, I can subtract the exponents. So it's going to be (-70 - 28) which is (-98). And so this is going to simplify (2^{-98}) power, and that's another way of viewing the same expression.