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We're asked to simplify m to the third and then that

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squared all over mx to the third all of that to the

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fourth power.

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And they want us to express our answer

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in exponential form.

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And then they tell us that m does not equal zero and x does

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not equal zero.

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Because if they did, then this denominator would be equal to

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zero if either of them did and then that

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would just be undefined.

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So that's what they have to put that little caveat there.

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That neither of them are going to be equal to zero.

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So let's try to simplify this.

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Let me rewrite it.

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So the numerator is m to the third squared.

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And our denominator is mx to the third to the fourth power.

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Now let's just work on each of these independently.

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What is m to the third and then that to the second power?

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Well if you raise something to an exponent, and then raise

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that whole thing to another exponent, this is going to be

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the same thing.

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So this numerator up here is going to be m to

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the 3 times 2 power.

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Or m to the sixth power.

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We're raising it to third and then squaring it.

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And if you think about it, this expression right here--

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let me just do a little aside here.

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M to the third squared, there's no magic behind why

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were multiplying that.

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If you think about m to the third squared, this is the

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same thing as m to the third times m to the third.

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And if you saw something like this, you would

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add these two exponents.

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3 plus 3.

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You have exactly two 3's here.

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If this was 4, you would be multiplying it four times.

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So you'd have four 3's.

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So that's why we're multiplying these two

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numbers, 3 times 2.

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So that gives us m to the sixth.

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And then the denominator here, we have mx to the third to the

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fourth power.

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Now in general, when you have the product of some numbers,

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and you're raising the entire product to some exponent, you

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can raise each of the terms to that exponent.

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So this denominator right here, this is going to be m to

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the fourth times x to the third to the fourth.

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I'm just raising both of these terms, m to the fourth x to

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the third, to the fourth power.

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Now, what does this simplify to?

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Well, our numerator is still m to the sixth.

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I'll do one step at a time.

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Our denominator is m to the fourth.

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But we have x to the third and then that to the fourth power.

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So we can multiply these.

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3 times 4.

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So this is x to the 12th power, so times x to the 12th.

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Now, we have an m to the sixth in the numerator and an m to

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the fourth in the denominator.

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You can view this as the same thing as m to the sixth times

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m to the negative four, either way.

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But we know, or hopefully we know, that this right here can

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be simplified as m to the sixth minus 4 power.

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So this is going to be equal to-- so the numerator, we

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could have m to the sixth minus 4, which is m squared.

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That's what this simplifies to.

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And the denominator we still have an x to the 12th.

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And we're done.

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If we don't want something in the denominator, we could

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rewrite this x to the 12th in the denominator as x to the

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negative 12th in the numerator.

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So we could rewrite this whole expression as being equal to m

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squared times x to the negative 12.

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Either of these would be an acceptable answer.

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