### Video Transcript

Evaluate the expression 64 to the power of negative a third over one over 625 to the power of negative a half all to the power of negative a half.

So whenever we’re looking at a question like this, the first thing we want to do is look at the base numbers. And we want to think, “Is there a simpler base number that we can convert them into?” And we can do that using exponents or powers ourselves.

Well, to start off with, we can rewrite the numerator. It’s two to the power of six to the power of negative a third. And that’s because 64 is equal to two to the power of six. And we get that because if we know that two cubed is eight, then you can multiply eight by two; you get 16. So that will be two to the power of four. Multiply 16 by two gives you 32, which is two to the power of five. If you multiply 32 by two, you get 64, which is equal to two to the power of six. Okay, that’s our numerator. And then if we’re looking to rewrite the denominator as well, we can write this as one over five to the power of four all to the power of negative a half. And that’s because 625 is equal to five to the power of four.

So now, what we’re gonna look at is a couple of exponent rules because what they can help us do is evaluate our expression. So first of all, we know that 𝑥 to the power of negative 𝑎 is equal to one over 𝑥 to the power of 𝑎. And we also know that 𝑥 to the power of 𝑎 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏. So, you multiply the exponents.

So, the first thing we’re gonna do is simplify the denominator because we’re gonna apply the first rule we looked at. And if we do, what we can do is rewrite one over five to the power of four as five to the power of negative four. So, now, what we can do is simplify both the numerator and denominator. And we’re gonna do that using the second rule that we just mentioned. And if we do that, what we’re gonna get is two to the power of negative two as the numerator. And that’s because if you have six multiplied by negative a third, you get negative two. Then, this is to the power of negative a half. And the reason we can do that is because if you’ve got the whole thing, so the whole fraction to the power of negative a half is the same as taking the power of negative a half of both the numerator and denominator.

So then if we look at the denominator, we’ve got five squared to the power of negative a half. And that’s because we had five to the power of negative four to the power of negative a half. So if you multiply negative four by negative a half, negative multiplied by negative is positive. So, we get two. And now if we apply our rule again, what we’re gonna have is two to the power of one as the numerator. I’ve written it here in orange just because you wouldn’t usually write the one.

But if you had negative two multiplied by negative a half, this is gonna give positive one, so just showing that that’s the result you would get. However, you would just write it as two over five to the power of negative one. And that’s because if you got two multiplied by negative a half, it’s negative one. Well, if we’ve got two over five to the power of negative one, well, five to the power of negative one is the same as one over five or one-fifth. So therefore, it’s the same as two divided by a fifth.

So now, what we can do is use our memory aid to remind us how we would divide by a fraction. And that is KCF: Keep it, Change it, Flip it. So, what we do is we keep the first number. So that’s just two. Then, we change the sign so that’s become multiply not a divide. Then we flip our fraction, so it’s the reciprocal. So instead of one over five, we get five over one. So, we’ve got two multiplied by five over one. So, this is gonna give us 10.

So, we can say that if we evaluate the expression 64 to the power of negative a third over one over 625 to the power of negative a half all to the power of negative a half, the result is 10.