Rewriting expressions with exponents challenge 2 | Algebra 1 (TX TEKS) | Khan Academy

So we have an expression here that has a bunch of exponents in it. It seems kind of complicated, and what I want you to do, like always, is pause this video and see if you can work through this yourself. Essentially, working through this means simplifying it on your own before we work through this together.

All right, now let's work through this together. So we have this expression times this expression, but then this expression is raised to an exponent. So, order of operations would tell me, "Hey, let's do the exponent first before we actually multiply things."

And how do we simplify this? We have a bunch of stuff that is raised to an exponent. Well, we've seen that exponent property before. If I have A times B, and all of that is raised to the n-th power, that's the same thing as A to the n times B to the n.

So, let's rewrite this part like that. I'll just rewrite the first part, so we have 4A^3B. Then, I will do this part in blue just to make it look a little different. So, I'm raising all of this to the 1/2 power, so this is times 81 to the 1/2 power times A^2 to the 1/2 power times B to the E to the 1/2 power.

Let me write a little bit neater. All right, to the 1/2 power. And now what do we do with all of this? Well, actually, before we even get there, what's 81 to the 1/2 power? Well, that's what number times itself is 81. That, of course, is going to be 9. So we got that simplified.

What's A^2 to the 1/2 power? Well, there, we need to remind ourselves that, in purple, if I have A to the n and then I were to raise that to the m, that is equal to A to the n times m power. So we're just going to multiply these exponents here.

So, what is 2 times 1/2? Well, that's just going to be 1. So, actually, let me just rewrite all of this. It is going to be 4A^3B and then times I have 9. And then in purple, I have A^2 to the 2, which is A^(2 times 1/2). 2 times 1/2 is 1, so this is A to the first power. A to the first power is the same thing as just an A.

Then we will do that again right over here, B to the E and then I raise that to the 1/2, so I could just multiply these two exponents. 8 times 1/2 is 4, so this is the same thing as B to the 4th power. B times B to the 4th power, and I'll close my parenthesis.

Well, now I have a bunch of things being multiplied times a bunch of other things being multiplied. So one way to think about it is I could just remove the parentheses if I like. Why don't I do that? I'll just—oops, I thought I was erasing—let me just remove the parentheses if I like.

Now I'm just multiplying a bunch of things. It might not be written in the neatest way possible, but we can. I'll put a little multiplication here and if I'm just multiplying a bunch of things, I could change the order of that multiplication. So I'd like to multiply my coefficients first.

So, I'm going to have 4 times 9, which is going to be 36. And then that's going to be times—let me do this in the salmon color—so A^3 times A. Actually, let me just write that. I don't want to skip too many steps, so times A^3 times A.

And then last but not least, let's get to—we have B times B to the 4th, times B times B to the 4th. And what I'm doing here is I'm just changing the order of multiplication, so I'm multiplying parts of the expression that have the same base because now I can use more exponent properties.

We know, or we have seen before, if I have A to the n times A to the m, this is equal to A to the n plus m. Same base, then I can add the exponents. So this right over here is going to be A to the 3 plus 1 power, right over here.

So all of this is A to the 3 plus 1, and all of this—this is B to the 1. That's the same thing as B, so this is going to be the same thing as B to the 1 plus 4 power. And we are in the home stretch.

This is all going to simplify to 36 times A to the 4th power, A^4, and then B to the 1 plus 4 power is B to the 5th. And we are done.