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Simplify 25 a to the third

2
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and a to the third is being raised to the third power, times b squared

3
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and all of that over 5 a squared, b times b squared

4
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So we can do this in multiple ways, simplify different parts.

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What I want to do is simplify this part right over here.

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a to the third power, and we're raising that to the third power.

7
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So this is going to be from the power property of exponents, or the power rule

8
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this is going to be the same thing as a to the 3 times 3 power

9
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So this over here (let me scroll up a little bit)

10
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is going to be equal to a the 3 times 3 power, or a to the ninth power.

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We could also simplify this b times b squared over here.

12
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This b times b squared, that is the same thing as b to the first power

13
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remember, b is just b to the first power. So it's b to the first power times

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b to the second power. So b to the first times b to the second power

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is just equal to b to the one plus two power, which is equal to b to the third power

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and then last thing we could simplify, just right off the bat just looking at this:

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we have a 25 divided by 5. Well that's just going to give us 5.

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or we could say it's going to give us 5 over 1

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if you view it as dividing the numerator and the denominator both by 5.

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So what does our expression simiply to?

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We have 5a to the ninth, and then we still have this b squared here, b squared.

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All of that over a squared times b to the third power... times b to the third power.

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Now, we can use the quotient property of exponents.

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You have an a to the ninth

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Let me use a slightly different color. We have an a to the ninth.

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over a squared. What's that going to simplify to?

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Well, that's going to simplify to be the same thing,

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let me write this

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a to the ninth over a squared, the same thing as a to the nine minus two, which is equal to

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a to the seventh power.

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Now we also have

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and this will get a little bit interesting here

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We have a b squared over b to the third power

34
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So that simplifies too.

35
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So b squared over b to the third is equal to

36
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b to the two minus three power, which is equal to

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b to the negative one power.

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And we'll leave it alone like that right now.

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So this whole expression simplifies to

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It simplifies to: 5 times a to the seventh power (because this simplifies to a to the seventh)

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a to the seventh, times (the bs right here simplify too)

42
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b to the negative one.

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We could leave it like that, you know, that's pretty simple

44
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but we may not want a negative exponent there

45
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we just have to remeber that b to the negative one power is the same thing as

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one over b. Now if we remember that, then we can rewrite this

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entire expression as, the numerator will have a five

48
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and will have a to the seventh

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5 a to the seventh. And then the denominator will have the b.

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So we're multiplying this times one over b.

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That's the same thing as b to the negative one.

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And we are done!