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- [Presenter] So we have
this pretty complicated,

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some would say hairy,
expression right over here.

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And what I want you to
do is pause this video

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and see if you can simplify this

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based on what you know
about exponent rules.

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All right, now let's do this together.

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There's many ways you could approach this,

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but what my brain wants to do is first

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try to simplify this part right over here.

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I have a bunch of stuff in
here to an exponent power.

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And one way to think about
that, if I have, let's say,

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A times B

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to the, let's call it C, power.

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This is the same thing
as A to the C times B

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to the C power.

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So we could do that with
this part right over here.

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And actually, let me just simplify this

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so I don't have to keep rewriting things.

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So this can be rewritten as

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5M, or let me be careful.

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This is gonna be five squared

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times M to the negative one third squared

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times N squared,

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which is the same thing as 25.

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Now if I raise something to an exponent

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and then raise that to an exponent,

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so there's another exponent property here.

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If I have A to the B, and
then I raise that to the C,

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then I multiplied the exponents.

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This is equal to A to the B times C power.

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So here we would multiply these exponents.

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So it's 25M.

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Two times negative one third
is negative two thirds.

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And then of course we have
this N squared right over here.

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So actually lemme just rewrite everything

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so we don't lose too much track.

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So we have 75.

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I wrote M. 75.

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M to the one third. N
to the negative seven.

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And then I simplified the bottom part.

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I'll do that same color. As 25.

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M to the negative two thirds

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N squared.

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Now, some of y'all might
immediately be able

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to skip some steps here, but I'll try

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to make it very, very explicit.

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What I'm going to read, what I'm gonna do

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is rewrite this expression
as the product of fractions

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or as a product of rational expression.

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So I could rewrite this.

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This is being equal to

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75 divided by 25,

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which I think you know what that is.

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But I'll just write it like that.

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Times, and then we'll worry
about these right over here.

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Times M to the one third

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over M to the negative two thirds,

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and then times, I'll put this in blue.

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Times N to the negative seven

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over N squared.

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Now 75 over 25, we know what that is,

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that's going to be equal to three.

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But how do we simplify
this right over here?

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Well, here we can remind ourselves

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another exponent property.

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If I have, let's call it A to the B

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over C to the D, actually it
has to have the same base.

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Over A to the C.

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This is going to be the same thing

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as A to the B minus C power.

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So I can rewrite all of this business.

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I have my three here.

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Three times

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M to the one third.

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And then I'm gonna subtract this exponent.

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We have to be very careful.
We're subtracting a negative.

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So we're subtracting negative two thirds.

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That's all that exponent for M.

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And then we're going to have times N

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to the negative seven power

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minus two.

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And so now we are in the home stretch.

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This is going to be equal to

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three times M to the,

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what's one third minus
negative two thirds?

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Well, that's the same thing
as one third plus two thirds,

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which is just three
thirds, which is just one.

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So this is just M to the first power,

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which is the same thing as just M.

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And then that is going to be times

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negative seven minus two,
that is negative nine.

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So times N to the negative
ninth power. And we are done.

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And that is strangely
satisfying to take something

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that hairy and make it,
I guess, less hairy?

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Now, some folks might not like

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having a negative nine exponent here.

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They might want only positive exponents.

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So you could actually rewrite
this and we could debate

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whether it's actually
simpler or less simple.

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But we also know the exponent properties

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that if I have A to the negative N,

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that is the same thing
as one over A to the N.

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So based on that, I could
also rewrite this as three.

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We do the same color as that three.

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As three times M.

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And then instead of saying
times N to the negative nine,

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we could say that is over,

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that is over, N to the ninth.

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So that's another way to
rewrite that expression.