Rewriting expressions with exponents challenge 2
About this video
This video simplifies (4a^3 b)(81 a^2 b^8)^(1/2) step by step. It uses the power of a product rule, the power of a power rule (multiply exponents) and the product rule for the same base (add exponents) to reach 36a^4 b^5.
You'll be able to
- apply a fractional exponent to every factor in a product, using (ab)^n = a^n b^n.
- simplify a power of a power by multiplying exponents, such as (b^8)^(1/2) = b^4.
- reorder a product and add exponents of the same base to simplify an expression.
Key words
- Exponent
- The small number that tells you how many times a base is used as a factor, or which root/power to take.
- Base
- The number or variable that is being raised to an exponent, like a in a^3.
- Coefficient
- The number multiplying the variables in a term, like 4 in 4a^3 b.
- Order of operations
- The agreed order for doing calculations, where exponents are done before multiplication.
- 1/2 power
- Raising a number to the 1/2 power means finding the number that times itself gives it, so 81^(1/2) = 9.
Transcript
- [Instructor] So we have an expression here that has a bunch of exponents in it and it seems kind of complicated. And what I want you to do, like always, 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 where 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 nth 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 to the third b, and 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 squared to the 1/2 power times b to the eighth 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 squared to the 1/2 power? Well, there we need to remind ourselves, I'll do that in purple, that 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 gonna multiply these exponents here. So what is 2 times 1/2? Well, that's just going to be one. So actually let me just rewrite all of this. This is going to be 4a to the third b, and then times, I have nine.
And then in purple, I have a squared to the 1/2, which is a to the 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. And then we will do that again right over here. b to the eighth, 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 fourth power. b times b to the fourth power. And I'll close my parentheses. 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, I could just remove the parentheses, if I'd like. Why don't I do that? I'll just, whoops, I thought I was erasing. I could 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 to the third times a, actually let me just write that. I don't wanna skip too many steps. So times a to the third times a. And then last but not least, let's get to the, 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, 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 is, this is b to the first. 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 3 plus 1, is a to the fourth power, a to the fourth. And then b to the 1 plus 4 power is b to the fifth. And we are done.
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