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# Simplifying quotient of powers (rational exponents)

Video transcript

- [Voiceover] So we have an
interesting equation here. Let's see if we can solve for k, and we're going to assume
that m is greater than zero. Like always, pause the video. Try it out on your own, and
then I will do it with you. All right, let's work
on this a little bit. So you can imagine that the key to this is to simplify it using our knowledge of exponent properties, and there's a couple of
ways to think about it. First, we can look at this
rational expression here, m to the 7/9 power divided by m to the 1/3 power. And the key realization here is that if I have x to the a over x to the b, that
this is going to be equal to x to the a minus b power. And actually comes
straight out of the notion that x to the a over x to the b, x to the a over x to the b, is the same thing as x to the a times one over x to the b, which is the same thing
as x to the a times... One over x to the b, that's the same thing as x to the negative b, which is going to be the same thing as... If I have a base to one exponent times the same base to another exponent, that's the same thing as that base to the sum of the
exponents, a plus negative b which is just gonna be a minus b. So, we got to the same place. So, we can re-write this as... So, we can re-write this
part as being equal to m to the 7/9 power minus 1/3 power is equal to, is equal to m to the k over nine. And I think you see where this is going. What is 7/9 minus 1/3? Well, 1/3 is the same thing, if we want to have a common denominator, 1/3 is the same thing as 3/9. So, I can re-write this as 3/9. So 7/9 minus 3/9 is going to be 4/9. So, this is the same thing as m to the... M to the 4/9 power is going to be equal to m to the k-ninths power. So, 4/9 must be the
same thing as k-ninths. So, we can say 4/9 is equal to k-ninths. Four over nine is equal to k over nine, which tells us that k must be equal to four, and we're all done.