Simplify 16𝑎 to the power of six over eight 𝑎 to the power of negative nine, leaving your answer in exponential form.
First of all, just to make it easy, let’s show what we’re gonna be doing. I want to actually split our fraction up. And I want to split it up into numbers and then terms involving 𝑎. If we think about how a fraction works, we’ve got 𝑎𝑏 over 𝑐𝑑 is the same as 𝑎 over 𝑐 multiplied by 𝑏 over 𝑑 because then, obviously, when we’re multiplying fractions, we just multiply the numerators and multiply the denominators.
And by using this relationship, we’re left with- this is equal to 16 over eight multiplied by 𝑎 to the power of six over 𝑎 to the power of negative nine. And this is gonna be really useful for me now to show you how we’d actually simplify. Well, for our first term, we now- we’ve just got two because 16 divided by eight is just two. And this is gonna be multiplied by 𝑎 to the power of six divided by 𝑎 to the power of negative nine.
I’ve just rewritten it like this. So it looks more like the exponent rule that we’re gonna have a look at now just to help you understand. You don’t necessarily need to step here when you’re working it out. Well, the exponent rule that I’m gonna use to actually help me simplify this fully is this one here. And it states that 𝑎 to the power of 𝑚 divided by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 minus 𝑛.
So if we apply this to our expression, we get two multiplied by. And then we’ve got inside the parentheses 𝑎 to the power of six minus negative nine because six is our 𝑚 and the negative nine will be our 𝑛 in the exponent rule. Okay, great! So now we can take this to the final stage to fully simplify.
So we can actually say that if we simplify 16𝑎 to the power of six over eight 𝑎 to the power of negative nine, leaving the answer in the exponential form, our final answer will be two 𝑎 to the power of 15. And like I said before, we get 𝑎 to the power of 15 because it’s 𝑎 to the power of six minus negative nine. So that’s the same as six add nine, so two 𝑎 to the power of 15.