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In this video, I want to do a bunch of examples involving

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exponent properties.

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But, before I even do that, let's have a little bit of a

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review of what an exponent even is.

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So let's say I had 2 to the third power.

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You might be tempted to say, oh is that 6?

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And I would say no, it is not 6.

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This means 2 times itself, three times.

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So this is going to be equal to 2 times 2 times 2, which is

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equal to 2 times 2 is 4.

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4 times 2 is equal to 8.

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If I were to ask you what 3 to the second power is, or 3

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squared, this is equal to 3 times itself two times.

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This is equal to 3 times 3.

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Which is equal to 9.

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Let's do one more of these.

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I think you're getting the general sense, if you've never

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seen these before.

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Let's say I have 5 to the seventh power.

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That's equal to 5 times itself, seven times.

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5 times 5 times 5 times 5 times 5 times 5 times 5.

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That's seven, right?

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One, two, three, four, five, six, seven.

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This is going to be a really, really, really, really, large

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number and I'm not going to calculate it right now.

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If you want to do it by hand, feel free to do so.

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Or use a calculator, but this is a really, really, really,

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large number.

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So one thing that you might appreciate very quickly is

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that exponents increase very rapidly.

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5 to the 17th would be even a way, way more massive number.

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But anyway, that's a review of exponents.

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Let's get a little bit steeped in algebra, using exponents.

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So what would 3x-- let me do this in a different color--

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what would 3x times 3x times 3x be?

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Well, one thing you need to remember about multiplication

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is, it doesn't matter what order you do the

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multiplication in.

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So this is going to be the same thing as 3 times 3 times

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3 times x times x times x.

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And just based on what we reviewed just here, that part

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right there, 3 times 3, three times, that's 3

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to the third power.

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And this right here, x times itself three times.

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that's x to the third power.

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So this whole thing can be rewritten as 3 to the third

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times x to the third.

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Or if you know what 3 to the third is, this is 9 times 3,

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which is 27.

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This is 27 x to the third power.

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Now you might have said, hey, wasn't 3x times 3x times 3x.

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Wasn't that 3x to the third power?

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Right?

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You're multiplying 3x times itself three times.

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And I would say, yes it is.

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So this, right here, you could interpret that as 3x to the

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third power.

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And just like that, we stumbled on one of our

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exponent properties.

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Notice this.

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When I have something times something, and the whole thing

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is to the third power, that equals each of those things to

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the third power times each other.

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So 3x to the third is the same thing is 3 to the third times

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x to the third, which is 27 to the third power.

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Let's do a couple more examples.

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What if I were to ask you what 6 to the third times 6 to the

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sixth power is?

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And this is going to be a really huge number, but I want

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to write it as a power of 6.

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Let me write the 6 to the sixth in a different color.

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6 to the third times 6 to the sixth power, what is this

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going to be equal to?

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Well, 6 to the third, we know that's 6 times

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itself three times.

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So it's 6 times 6 times 6.

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And then that's going to be times-- the times here is in

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green, so I'll do it in green.

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Maybe I'll make both of them in orange.

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That is going to be times 6 to the sixth power.

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Well, what's 6 to the sixth power?

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That's 6 times itself six times.

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So, it's 6 times 6 times 6 times 6 times 6.

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Then you get one more, times 6.

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So what is this whole number going to be?

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Well, this whole thing-- we're multiplying 6 times itself--

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how many times?

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One, two, three, four, five, six, seven,

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eight, nine times, right?

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Three times here and then another six times here.

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So we're multiplying 6 times itself nine times.

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3 plus 6.

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So this is equal to 6 to the 3 plus 6 power or 6

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to the ninth power.

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And just like that, we/ve stumbled on

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another exponent property.

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When we take exponents, in this case, 6 to the third, the

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number 6 is the base.

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We're taking the base to the exponent of 3.

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When you have the same base, and you're multiplying two

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exponents with the same base, you can add the exponents.

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Let me do several more examples of this.

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Let's do it in magenta.

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Let's say I had 2 squared times 2 to the fourth times 2

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to the sixth.

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Well, I have the same base in all of these, so

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I can add the exponents.

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This is going to be equal to 2 to the 2 plus 4 plus 6, which

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is equal to 2 to the 12th power.

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And hopefully that makes sense, because this is going

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to be 2 times itself two times, 2 times itself four

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times, 2 times itself six times.

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When you multiply them all out, it's going to be 2 times

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itself, 12 times or 2 to the 12th power.

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Let's do it in a little bit more abstract way, using some

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variables, but it's the same exact idea.

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What is x to the squared or x squared times x to the fourth?

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Well, we could use the property we just learned.

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We have the exact same base, x.

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So it's going to be x to the 2 plus 4 power.

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It's going to be x to the sixth power.

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And if you don't believe me, what is x squared?

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x squared is equal to x times x.

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And if you were going to multiply that times x to the

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fourth, you're multiplying it by x times itself four times.

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x times x times x times x.

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So how many times are you now multiplying x by itself?

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Well, one, two, three, four, five, six times.

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x to the sixth power.

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Let's do another one of these.

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The more examples you see, I figure, the better.

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So let's do the other property, just to

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mix and match it.

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Let's say I have a to the third to the fourth power.

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So I'll tell you the property here, and I'll show you why it

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makes sense.

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When you add something to an exponent, and then you raise

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that to an exponent, you can multiply the exponents.

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So this is going to be a to the 3 times 4 power or a to

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the 12th power.

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And why does that make sense?

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Well this right here is a to the third

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times itself four times.

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So this is equal to a to the third times a to the third

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times a to the third times a to the third.

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Well, we have the same base, so we can add the exponents.

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So there's going to be a to the 3 times 4, right?

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This is equal to a to the 3 plus 33 plus three plus 3

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power, which is the same thing is a the 3 times 4 power or a

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to the 12th power.

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So just to review the properties we've learned so

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far in this video, besides just a review of what an

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exponent is, if I have x to the a power times x to the b

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power, this is going to be equal to x to

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the a plus b power.

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We saw that right here.

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x squared times x to the fourth is equal to x to the

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sixth, 2 plus 4.

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We also saw that if I have x times y to the a power, this

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is the same thing is x to the a power

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times y to the a power.

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We saw that early on in this video.

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We saw that over here.

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3x to the third is the same thing as 3 to the third times

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x to the third.

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That's what this is saying right here.

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3x to the third is the same thing is 3 to the third times

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x to the third.

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And then the last property, which we just stumbled upon

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is, if you have x to the a and then you raise that to the bth

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power, that's equal to x to the a times b.

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And we saw that right there. a to the third and then raise

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that to the fourth power is the same thing is a to the 3

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times 4 or a to the 12th power.

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So let's use these properties to do a handful of more

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complex problems. Let's say we have 2xy squared times

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negative x squared y squared times

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three x squared y squared.

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And we wanted to simplify this.

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This you can view as negative 1 times x

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squared times y squared.

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So if we take this whole thing to the squared power, this is

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like raising each of these to the second power.

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So this part right here could be simplified as negative 1

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squared times x squared squared, times y squared.

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And then if we were to simplify that, negative 1

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squared is just 1, x squared squared-- remember you can

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just multiply the exponents-- so that's going to be x to the

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fourth y squared.

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That's what this middle part simplifies to.

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And let's see if we can merge it with the other parts.

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The other parts, just to remember, were 2 xy squared,

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and then 3x squared y squared.

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Well now we're just going ahead and just straight up

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multiplying everything.

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And we learned in multiplication that it doesn't

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matter which order you multiply things in.

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So I can just rearrange.

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We're just going and multiplying 2 times x times y

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squared times x to the fourth times y squared times 3 times

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x squared times y squared.

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So I can rearrange this, and I will rearrange it so that it's

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in a way that's easy to simplify.

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So I can multiply 2 times 3, and then I can worry about the

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x terms.

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Let me do it in this color.

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Then I have times x times x to the fourth times x squared.

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And then I have to worry about the y terms, times y squared

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times another y squared times another y squared.

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And now what are these equal to?

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Well, 2 times 3.

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You knew how to do that.

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That's equal to 6.

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And what is x times x to the fourth times x squared.

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Well, one thing to remember is x is the same thing as x to

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the first power.

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Anything to the first power is just that number.

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So you know, 2 to the first power is just 2.

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3 to the first power is just 3.

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So what is this going to be equal to?

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This is going to be equal to-- we have the same base, x.

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We can add the exponents, x to the 1 plus 4 plus 2 power, and

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I'll add it in the next step.

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And then on the y's, this is times y to the 2

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plus 2 plus 2 power.

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And what does that give us?

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That gives us 6 x to the seventh power, y

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to the sixth power.

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And I'll just leave you with some thing that you might

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already know, but it's pretty interesting.

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And that's the question of what happens when you take

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something to the zeroth power?

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So if I say 7 to the zeroth power, What does that equal?

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And I'll tell you right now-- and this might seem very

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counterintuitive-- this is equal to 1, or 1 to the zeroth

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power is also equal to 1.

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Anything that the zeroth power, any non-zero number to

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the zero power is going to be equal to 1.

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And just to give you a little bit of

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intuition on why that is.

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Think about it this way.

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3 to the first power-- let me write the powers-- 3 to the

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first, second, third.

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We'll just do it the with the number 3.

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So 3 to the first power is 3.

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I think that makes sense.

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3 to the second power is 9.

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3 to the third power is 27.

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And of course, we're trying to figure out what should 3 to

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the zeroth power be?

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Well, think about it.

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Every time you decrement the exponent.

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Every time you take the exponent down by 1, you are

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dividing by 3.

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To go from 27 to 9, you divide by 3.

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To go from 9 to 3, you divide by 3.

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So to go from this exponent to that exponent, maybe we should

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divide by 3 again.

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And that's why, anything to the zeroth power, in this

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case, 3 to the zeroth power is 1.

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See you in the next video.