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Powers Of Monomials
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Apr 25, 2025
In this section ( 9.6), we'll learn how to apply the power a power when it comes monomials. We'll also learn how to apply the power of a product and then close the section with a life application problem. Chapters:
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0:00
so we are talking about powers of a
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monomial I've already defined what a
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monomial is right a monomial is a
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product of um variable uh or a constant
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So we talked about this here like for
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example 3x that's a monomial 3 is a
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monomial 5 by itself would be a monomial
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5xy that would be a monomial because it
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is a combination of a um a variable and
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a constant Just a constant by itself or
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just a variable Right but as long as
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nothing separate them like here this
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will not qualify as a monomial because
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it's no longer just one term It's a it's
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your adding right so we defined that in
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uh earlier on in chapter 9 Now today
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we're going to talk about powers We've
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also done a lot of things with powers
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But the one thing that you need to learn
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now is how do you compute the power of a
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power right how do you compute the power
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of a power so that's what we going to
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learn Now this is a monomial here right
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i have x and x is raised to the power of
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a and then the whole thing is raised to
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the power of b So how do you compute
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that how do you evaluate that that's
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what we're going to learn in this
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chapter Okay Now this is just symbols
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I'm just telling you what it is Now I'm
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actually going to give you an actual
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example For example I have three to
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three raised to the power of two The
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whole thing is raised to the power of
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five And I'm asking you to evaluate that
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So how would you do it so this is 3^ squ
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raised to the^ 5 So how do we find that
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out so to do that we're going to use the
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rule that we just learned about x raised
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to the power of a the whole thing raised
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to the power b So how do you do it you
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multiply the exponent That's all you're
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doing All you have to do here is
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multiply the exponent right so 3 to the
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two the whole thing raised to the 5 is
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equal to 3 to the 10 because 2 * 5 is 10
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Okay So how come you guys have nothing
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that's right about you guys is just
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sting and then later on you going to
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tell me that we don't understand this
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Okay It doesn't matter We still have to
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write this down You don't want to do it
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then don't come to school So that's the
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only go to jail All right So this is 3
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to the 10 Okay So I would be able So
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basically this is called the power of a
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power right So 3 raised to the two for
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example if I have two raised to the 5
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the whole thing raised to the six and I
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want you to evaluate this So how would
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you do this Gabriel
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oh you do like
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you do like two
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So what did I say i just mentioned the
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rule here Were you
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listening right Did you see what I did
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here right Uh 3 to the two the whole
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thing raised to the 5 So what do you do
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to the exponent
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it's on the board
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You just adding I mean you multiply You
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multiply it right so that therefore how
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would you do this here 2 5 the whole
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thing to the six
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30 2 30 That's right This is 2 to 30
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because all you're doing is you're
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multiplying the exponent right so this
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is how you do this So 2 to 5 the whole
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thing to the 6 is going to give you 2 to
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the 30 Now let's try something else here
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What if I have x to the 4 the whole
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thing raised to the 5 So what would that
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give
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you x to the 4 the whole thing raised to
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the five What does that give us
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x 20 X to the 20 right because all we're
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doing is multiplying the exponent 4 * 5
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that is 20 so it's X to the 20 okay all
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right now the next one I
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have 4 to the
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3 raised to the 7 and I want to evaluate
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this so what's that going to give me
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4 to the 21 4 to the 21 does everybody
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know how we did that Kenzie you get
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43 this is seven Oh yeah that's four and
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then exponent 21 21 So all you're doing
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is multiplying the exponent Again this
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is called the power of a power Right so
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this is how you do that Now what about y
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4 the whole thing to the four and I want
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you to find this y to the 16 Y to the 16
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right y to the 16 because you're
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multiplying this Now we have something
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with a negative exponent I have
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23 raised to the -2 Uh Kenzie what's
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that going to give you 23rd raised to
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the -2 Yeah
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Oh 26 26 And remember what we did in the
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very like two two sections ago when we
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have a negative exponent How do you
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write it by getting rid of the negative
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what do we do you remember who remembers
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how to do this when you have a negative
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exponent
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anybody wants extra credit do it
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the other way How do you do it you take
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it and you put like one over 2 to the 6
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That's right So that becomes one over
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what one over 2 to the 6 right which is
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basically 1 over 64 right so anytime you
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have a negative exponent you have to
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flip it You never keep a negative
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exponent You always have to turn it into
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a positive exponent So that becomes 1 /
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26 which is 1 over 64 Right so that is
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how you do So I'm going to put twice
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here so I know that So that's extra
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credit right there So I want you to
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remember these things Okay all right So
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that's called a power of a power So let
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me give you another example What if I
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have x to the -3 the whole thing to the
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30 what would I give you x to the -3 the
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whole thing to 30 X to the positive 30
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Right Do I need to make any changes here
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no No What if I have
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y^2 the whole thing to the^ 7 what do we
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get
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y^4 And then what
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1 y to the 14 14 Not negative right
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positive because we want to get rid of
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the negative Anytime you have a negative
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exponent you want to make sure you turn
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that negative exponent into a positive
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by flipping it right according to the
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rule that we learned in the last two
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sections All right So that's called a
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power of a power Now now we're going to
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stretch it a little bit Now we have what
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you call the power of a product right
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the power of a product And what does
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that mean here i don't just have one uh
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expression I have xy right xy and xy is
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raised to the power of m Okay So when
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that happens each entry is going to be
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raised to the power of m What do I mean
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by that so if I have here 5 x to the 7
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right 5x the whole thing to the 7
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because the expression is in parenthesis
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That means five is going to be raised to
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the seven and x is also going to be
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raised to the seven Does that make sense
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right i have two not two things here I
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have a five which is the constant and I
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have the variable x and the whole thing
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is raised to the power of seven So this
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becomes 5 to the 7 * x to the 7 and the
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rule is called the power of a product
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Okay First thing is this We have the
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power of the power and now we have the
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power of a product So you have to do it
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uh for each of those um pieces in a
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polomial So example I have 3x the
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3r right 3x the 3r is raised to the
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power 4 So if you do it what's going to
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happen let me do it here I have 3x the
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3r and now it's raised to the power four
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So what's going to happen here
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what's that going to give me
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mhm Yeah Go ahead raise everything to
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the power So tell me what you're going
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to get Uh 3^ 4 *
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x^ 3 * 3 * 4 right because x has a power
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of three and based on what we learned
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earlier because this is a power you have
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to multiply the exponents right so that
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gives you 3 to the 4 and then x to the
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12 Now when you have numbers like this
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like 3 to the 4 we know we can find what
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it is 3 to the 4th is what 3 * 3 * 3 * 3
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that gives you what 81 Okay so this is
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81 x to the 12 Okay so this is how you
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use the power of a product Now I gave
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you a couple of things here that I want
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I want us to try together Okay so let's
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do them one at a time Let's do them one
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at a time So the first one is uh we have
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6
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w 4 raised to the^ 5 So go ahead and try
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it on your own and tell me what you got
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So we have 6 w^ 4 raised to the^
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5 Let me see how you going to apply that
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uh product rule
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63^ 5
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Are we done no 30 more seconds
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[Music]
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done
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So all right So expand So what did you
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get how did you do it yeah So I did
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the power of product thing and I got 6^
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5 * W^ 20 That's good That's good And
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then what and now we can find 6 to the^
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5 right So 6 to the^ 5 what did I give
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you i got 7,776
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7,776 All right So this is
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7,776 w to the power of 20 Okay Now
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let's try the next one I have -4 Let me
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write down here
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-4 x 5 y 7 the whole thing to the power
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of three Let's see what you get on there
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It's like it's just okay Yeah
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So4 x 5 y to the 7 and the whole
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expression is raised to the power of
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three
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All right let's see 45 more seconds
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Have you done so what did you get so
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tell me what you So go step by step Uh I
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added -4
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Okay And added x to the power of 15 15
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Mhm And and y^ 21 All right Does anybody
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agree with that right It's -4 to the 3
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power X to the 15 because 5 * 3 is 15 y
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to the 21 because 7 * 3 Now what do we
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do with4 to the third power
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do it You do the conversion You do the
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conversion thing One over Do we do the
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conversion here oh you know you just put
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it in the format Format So what's44 * 4
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that's 4 What does that give you 48 48
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Are you sure 16 * 4 that's 64 right So
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that's 64 And it's -64 X15 and then Y21
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right so it's -64 X15 Y21 because this
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is an odd number This is negative So
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you're going to get a negative right so
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it's -64 X15 Y21 Yes So so when you when
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you have like a negative exponent and
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you put that and you put that in that
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format that we did a while ago do you do
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the whole conversion or do you just put
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it in that format you just put in that
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format Let's do it on that the next one
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The next one has that right let's do
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that for the next next uh next problem
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can't raise something to a negative
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power So we have 3x^ -2 right y^ 4 and
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then the whole thing is raised to the
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power two So let's see what that gives
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us 3x^ -2 y^ 4 and then the whole thing
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is raised to the power of 2
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That was the
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So far so good But you do you do one
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more translation You're almost done
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All right Are we done yet just give me a
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minute All right A minute I give you 20
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seconds
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I don't know how to solve this I think I
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got it right I'm not sure All right So
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tell me what you got This looks like um
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I got three
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Three to the two Okay Then what times x
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and4 Okay And then y eight
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All right So so far so good Then what
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else we need to do here um to the
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Mhm
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Yeah It's nine Okay 9 Mhm And then I did
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the inversion cuz it's like negative
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inversion Okay And then times y 1 x 4
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like this right y All right Now I mean
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technically this is right but we can we
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have to go a step further right because
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this is 9 we can multiply what 9 * 1 * y
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8 Okay So because 9 is like 9 over 1
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Okay So that gives us what 9 y 8 over x
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4 Okay So this is in the denominator So
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you're going to 9
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* 1 * y 8 So this is 9 y over x 4 Okay
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So this is how you do this here Okay
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Okay Now we're going to we're going to
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finish and close this with a word
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problem You know what a cube is right
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how do you find the volume of a cube
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you do what
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you do what uh let's do it
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Okay let me write it again I'll write it
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again
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All right you got it It's time height So
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it's all right Now let me let me show
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you something here Right To find the
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volume of a cube the volume is pretty
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much s to the power three right s the^ 3
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S stands for what side right so side
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raised to the power 3 So because all the
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sides of the cube are the same they're
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congruent right they're all the same So
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therefore to find the volume of a cube
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you just take one side and you cube it
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Okay then you raise it to the power of
16:34
three This is the formula to find the
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volume of a cube Now I gave you the
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length of one side So now how are we
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going to apply what we learned today to
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find the volume of a cube so what are we
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going to do that length the length which
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is what
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two raised to the power of what
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three volume Well I thought you had you
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would have to to get the side you would
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have to multiply that by itself No this
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side this side all of them are the same
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They're all congruent Well side to power
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three Yes One panel of the side No no
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that's one side The panel just one side
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This side here right so you do length
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time width time height Well the sides
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are the same because it's a cube So
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therefore to find the volume of a cube
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you have to just raise the one side to
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the power three So what do we
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get if you expand this what's that going
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to give us oh uh 2 to the 3 x to the 3r
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y to the 15 y 15 and then go that's 8 x
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3r y to the 15th y to the 15 Right so
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this is how you apply this rule here So
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we learned two things here today the
17:50
power of a power and the the power of a
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product and then we apply it to word
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problem Okay want to find the volume of
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a cube You can use this formula to do it
17:59
You can simplify it because other people
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are going to go this time this time this
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too much work I'm just going to do it
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quickly I'm just going to raise it to
18:08
power three because our lengths are the
18:10
same That was such a word problem That
18:13
was such a good word problem That was
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such a good word problem All right So
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now I want to give you work