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Negative Exponents
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Apr 11, 2025
Learn everything you need to know about negative exponents in this fun and easy-to-follow pre-algebra lesson! We'll break down the concept of negative exponents, and show you how to simplify expressions like x⁻ⁿ step by step. With clear examples and helpful tips, you'll gain the confidence to tackle negative exponents like a pro. This video is perfect for students, parents, or anyone looking for a simple and engaging way to master this important math topic. Whether you're just starting out or need a quick refresher, we've got you covered!
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so
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basically when you have a negative
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exponent it is the standard that we
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don't keep it as a negative exponent We
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have to turn it into a positive And how
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do we do that We use this formula that I
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just gave you here Right If I have any
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number raised to the power of a negative
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you can turn it into a fraction Yes Why
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does
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um why does why does the formula work
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Why does it work
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Um why does it work
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That's a great question It works because
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it works right It works because they
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came up with formula and it works I
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don't know how to make I don't know how
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to understand You're asking the
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mechanism the mechanism behind it I
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don't I don't know how to understand the
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topic
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Well here's the thing The the pagonal
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theorem a square plus b square is equal
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to c square Right He studied it He
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understood he come came to the
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realization that this was always true
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Right is the same here X to the negative
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N is always going to be equal to 1 / X
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to the N So if you if you want to get a
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proof about it I'll probably look it up
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later for you and then I will explain it
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Uh but today we're just going to stick
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to understanding this is what it is
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right The y the y behind it We can go
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and then ask you we can elaborate on
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that later on So 4 to the -2 for example
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is 1 / 4 to the power of two Right Now
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the thing is this you do not leave it
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alone You have to change into a 4 square
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is 16 because 4 2 is 4 * 4 So therefore
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4 raised to the^ -2 is 1 / 4 2 which is
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1 16 right And I give you another
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example 3 to the 1 How do you turn this
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into a uh a positive exponent You you
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use the same um method that we use here
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1 3 to 1 which is 1 3
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for the
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number one exponent not negative example
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how would you turn this into a positive
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exponent one over two and then um two
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with the three to the power that's it
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which becomes what two to the three you
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know what 2 to the 3 power is what is it
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two eight so that's one over eight right
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so one over eight can be written as
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So basically whenever we do negative
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exponents we're going to get a fraction
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You got to get a fraction You got to get
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a fraction right What about m^4 How
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would you turn this into a positive
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exponent using a fraction Put one over m
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exponent four That's it Right What about
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3 to the 5 What does that equal 1 3
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which is basically 3 to the 5th power is
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243 Right So this is 1 over 243 Wait why
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is it 243 Is it shouldn't it be just 3
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five times 3 five times gives you 243
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Right So this gives you this Right And
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then what about y to the3 Well that
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equal
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one
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And what's two to the zero power
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One One Right Remember we said any
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number raised to the power of zero is
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always going to be it's one right So 2
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to the 0 is one No matter what number
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zero is equal to one right Unless the
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number under is not zero So zero to any
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number raised to the power of zero is
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one So that to 0 to the power something
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Yeah 0 to the power of anything is zero
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Zero is just zero right So only thing
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all the only thing that you need to
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retain here is this That is it So x to
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the n is equal to 1 / x to the n Right
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Good session Now the thing is this in
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this section you do never you never want
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to have like I remember in calculus
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anytime I got an answer with a negative
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exponent I have to turn into positive
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because my teacher will take some points
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off cuz it's just in mathematics there
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are certain things you just don't do
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They don't want that They don't want
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negative exponent They always want to
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get a positive exponent Right now we're
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gonna learn how to go back way Right If
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if how do I go from a fraction to a
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negative exponent I have 1 / 4 to the 2
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and I want to write each fraction as an
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expression using a negative exponent
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other than one So how do I turn these
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into a negative exponent with a base
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with a negative exponent Yes Um what
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would it just be Mhm Uh what two Huh
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would just be 4 to the^ of -2 That's
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right 4 to the -2 right Because this is
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equal to this So therefore this can go
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back to that That's what we're doing
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here right So 1 / 4 to the two we just
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to the -2 Right Now what about one over
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100 How do you do that
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One 100 It's just 100 right So we don't
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want the exponent to be what We don't
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want the exponent to be negative one We
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want it to be any other number but
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negative one So what can I do with 100
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How many times can get what can I do
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with 100 first 5 to the power of4 Uh 20
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No Oh no No How do I break down 100 You
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did remember we did a prime
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factorization or we did factorization
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like 100 can be broken down as what 15
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to 15 Yeah 15 10 and what 10 10 and 10
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to I can rewrite this as 10 to the^ of
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what Two right And then now can I change
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it
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Yeah We what 10 to the what
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102 That's it Right
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What about six to one over 6 to the 3
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What would that be -3 6
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What about one over 25
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That's five
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Because 25 is what
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Yeah 45 is five and five So this will be
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one over five to the two which is in
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turn is 5 to the -2 Right What about one
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over
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27 What's
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27 I want to have a base that has the
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same uh three Three and nine and nine is
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what Three and three So how many threes
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is that Three So one over 3 to the third
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power becomes what
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3 to the^ of3 Right So as you can see
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this this is really a sec like it's not
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that hard We just not done understanding
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mechanism Right now the next thing we're
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going to learn how to do and we're going
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to conclude this lesson is we're going
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to learn how to find evaluate algebraic
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expressions right with negative exponent
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For example if I have this here I want
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you to find
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4x to the -5 if x is equal to -2 So how
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do you how do you do this I want to find
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4x^2 if x=2 Yes do we get to solve for x
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We don't get to solve for x just yet
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We've done that before We've done that
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in the other chapters right So how do I
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evaluate this Evaluate When I say
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evaluate what am I asking for
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Just calculate it right So x is equal to
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what
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X=2 So we're going to replace it's going
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to be four * what to the power of
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now here right I have this four
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*25 it's it's not easy to do if I don't
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apply my uh transformation yes now what
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I do go from here so now we do 4 * -2
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and then now we can't we can't do 4 *2
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-2 is the power5 we can't do that right
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I thought one is part ofation ship Is
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this is this 45
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What is raised above
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Two -2 right Is
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4x5 the only number that's being raised
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to the 5 Yeah we just multiply both of
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them So it's8 to No no no You can't do
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that either right Because there's an
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exponent here So the first one -2 right
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So it's going to be 1 over what -2 to
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the^ 5 1 -2 5 This is what we need to do
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first right 4 * 1 -2 5th Now what is -2
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That's how many times Five
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right So it's -2 * -2 * -2 * -2 5 * that
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gives
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you 32 right So this by itself is -32
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That by itself
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right Because you do it five times You
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do -2 * -2 * -2 * -2 * we're going to
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get -32 So now this becomes 4 * 1 over
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-32 Now how do I multiply a number and a
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fraction I have to turn this into a what
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Four Four over what One right And then I
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just multiply across Right 4 * 1 is 4
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And then 1 * -32 is -32 Am I done or do
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I need to do something else here five At
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least what we know how to do that right
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Four goes into four four one time and
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then four goes into 32 eight times So
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this is one over8 right So this is how
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you're going to do that if you had uh to
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evaluate an expression Let's do a couple
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more of those evaluations of expression
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Right So say I
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have crap Say I have
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um I want you to find this here
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[Music]
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uh 2x^
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-4 if x is = 3 Right What am I going to
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do here
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2x to the4 Actually if x is equal to 4
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let's do four here So what am I going to
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do this one I want to find 2x to
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the4 What am I going to do here Uh
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replace x with four Replace x by four So
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we got 2 * what
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44
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right And now remember you can you can
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never go 2 * 4 is 8 You don't do that
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right Because 4 is raised to the power
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of4 There's an exponent here You can't
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multiply this two just yet You have to
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transform this first into a fraction
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before you do that Okay So that becomes
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what 2 * what
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One over four Four to the like that
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right Uh-huh And now this is two 1 / 4
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to the 4 4 to the 4 is 16 is 4 * 4 is
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going to be 4 * 4 That's 4 * 4 right 4 *
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16 * 16 which is I think 256 I think Uh
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put in your calculator I I don't I don't
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quite know Let me put it in my
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calculator Check my
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calculator I check on here What do you
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want me to do 16 * 16
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What's that goal
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16 * 16
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-240 Can't be negative Wait what What
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are we doing 16 * 16 Can somebody
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actually I did bro Somebody do it She's
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on a different dimension now Weird stuff
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Okay So she was right One right Which
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one Now guess what We have two So it's
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going to be 2 over 1 * that becomes 2
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over 256 And again you want to simplify
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right 2 goes into two one time and two
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go into 256 26 is 128 So it's 1
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over 128 Okay So this section is pretty
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much a wrap Right So now all we're going
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to do now is just do some