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How To Simplify Radicals
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Apr 1, 2025
In this section, we'll learn how to simplify radicals.
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is
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so we're talking
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about it's a new
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chapter
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called
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the and root right a lot of the time we
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we are used to like finding square
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root and this is pretty much what we
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doing like algebra one and maybe
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pre-alggebra some somehow now we're
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going to try to do higher are we using
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eyes are we using eyes we're going to be
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using at some point
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so now we have like square root right so
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when I say square root of four what I
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mean by that is this what
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number multiplied by itself twice gives
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you 64 what's the number 80 eight right
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so square root of 64 is eight so
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everybody knows that right that little
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three so when you have a square root
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what you're really saying is this is
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called an index the index is two but we
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don't write that it's not like it's it's
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we don't write that when you write
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square root of 64 you don't write the
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two there but there's a technically
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there's a two there right now to
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differentiate between this and that now
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this is called the cubic root right
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because it's a third so it's the cubic
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root okay so the question is this this
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reads cubic root of eight which means
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what number multiply by itself three
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times gives you eight two so the cube
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root of eight will be two so it's not
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such a hard section honestly if you
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understand the mechanism it makes it
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really easy right now we have the same
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question here this is called when you
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have a four to the fourth root right or
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so what's the fourth root of 81 three
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basically what number multiplied by
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itself four times gives you 81 is three
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if you want to break it down how would
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you do that how would you solve that so
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while you break down 81 right you can
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use the factor three a lot of people do
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that right 81 if you break down 81 that
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is what nine and nine nine and nine is
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what three and three and this is three
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and three right so 1 2 3 and four so
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this is basically three to the power of
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what four right so that means the fourth
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root of 33 4 is just three because
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because these are match you just drop it
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you get rid of the radical so you're
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left with three that's technically what
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we are doing in this section okay it's
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the same thing with this one here if I
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were to break down
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eight is four and two and we have two
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and two so this is basically cube root
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of two to the third power since these
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are matching you just drop the radical
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so you're left with what a single two
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right so that's what we're doing here
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You're just finding now I put n= n could
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be any number n could be 1 2 3 4 which
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we've done here or it could be six it
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could be any number you just have to
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make sure you match it up correctly do
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we do I have a question here or we good
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to continue any questions no all right
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now we have to do some examples so here
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I have plus or minus this is just a
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regular square root of 16 y 4 so what's
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that going to give you four y Plus orus
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4 y square 4 y square right because
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square root of y 4 is y square so you
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got to be careful with the understanding
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the the mechanism okay 16 is 4 of y^ 4
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is y^2 because here on the outside we
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have an imaginary two okay so this will
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be plus or - 4 y^2 right moving on to
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this one here actually I'll save this
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one last okay a little
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complicated so now I
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have x^2 - 8 to the^ 8 i want to find
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the square root what's that going to
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give me
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should you start with what's inside of
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the you start with x exponent is what we
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look at right it's easier so this is
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technically it's an imaginary two here
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right so an imaginary two so what do you
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think I'm going to get
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xus 4 it doesn't change nothing on the
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inside changes the only thing that I'm
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looking at is the exponent right so is
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it x - 8 * you mean x square - 8 x - 8
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to the power what fourth and then forget
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the
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negative right we are not looking at
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uh the inside expression does not affect
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the outcome because this is all raised
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to the power of eight right this is a
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square root so you mean that means you
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want it twice so this is to the four
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does that make sense any questions on
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that so if you if no number is written
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it's always just two since it's just a
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square root yeah mhm if there's no
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number you assume that this is a two so
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you just take out the square root before
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you solve it you uh I don't know if I
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understand what you're saying like
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explain that like are we going to solve
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it or we leaving it like that you're
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done when that's it you don't do
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anything else right because you want to
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get rid of the radical okay so here now
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we have what would you call this yeah
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i don't need to what would I want to
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simplify cuz you got to do four times
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it's going to be a huge number you want
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to do that probably not right cuz you
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going to do how many times this time
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that time that just going to be a huge
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number you just leave it alone don't
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make it any harder on yourself you can
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if you want but if she stuck here you
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may fight right not here we have a fight
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right now before I go on you got to be
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careful some people
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this right is not the same as this here
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right you see the five here is in the
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radical right here is outside this is
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five square root of 16 this is the fifth
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root of 16 right we have to be careful
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how we read this so you need to
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understand the distinction between the
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two okay now this is uh the fifth root
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of what 243 so what do I have to do to
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make it easier on
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myself first let's take it one at a time
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right these are going to be easy to
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break down aren't they right but then we
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need to look at
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uh 243 here right i'm going to break it
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down so that is what
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the number that goes in I think is three
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because if you do 2 + 4 is six 6 plus 3
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is nine when you add the digit of a
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number and if it's a multiple of three
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that means that number is divisible by
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three I don't know right so it be one
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you can do nine two three and n or nine
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whatever right so it be three and do I
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just find 81 right and this is basically
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nine and nine right this is three
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three so I got one two three four five
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three so I'm going to rewrite this as
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cube root of uh fifth root of 3 to the
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5 a to the 20 and b to the 25 now I can
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solve this right so that's going to give
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me what the first one is going to be
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three
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a to the four b to the 5 right so all
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you do is just you look at the exponent
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you divide it and whatever you get is
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whatever you get you get you get rid of
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the radical so this is 3 a to the 4th
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and then b to the 5th we're just making
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that simple as it can be that's it yeah
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making a simple by getting rid of the
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radical you're trying to like uh
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simplify that
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right all right now we have a different
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expression here now the problem is this
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one here is I have a negative under the
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radical so automatically we have to use
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what take I out i out there you go take
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I out good so you have a I out let's
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begin with I right here right nine
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what's of 16
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four four what's of x to the 4th 9 x1
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square and then square of y 8 y 4
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because this is an imaginary two here so
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do you see how this is actually really
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pretty simple stuff right it's not that
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complicated if you have once you
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understand the the gist of it it's just
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applying that if you have a square root
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you know that everything is all the
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exponent are divided by two if you have
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a fifth root five three by three so on
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and so forth now we're going to learn
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how to simplify using absolute value
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okay why I say that here Now if I was to
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simplify this
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right let's say you didn't know anything
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about this what would you get here i
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have 4 y to the 4 y y right but the
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danger is just putting y is
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this y could be a negative number
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couldn't it could be a negative number
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that means this would be a false
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statement right if y is a negative
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number this will be a false statement so
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therefore you have to put absolute value
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of y and let me explain why I'm saying
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the exponent of y is what here one one
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is odd because it's an odd number you
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can have a negative so anytime you take
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the you have an even number in even
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radical and you take the the root of it
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and you get a odd number a number with
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an odd um exponent you want to make sure
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you use your absolute value to avoid any
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danger of possibly saying that this
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number here is negative does that make
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sense can you just use absolute for all
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of them you can use absolute for all of
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them but you don't have to when you have
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this use absolute right for example here
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what would I give me if I was to do this
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here I have the six root of 64 right
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first I'm going to break down
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64 so I give me 8 and 8 right
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and eight is what four and two four and
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two that is two
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two two two i got one two three four
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five and six so this is like 6 2 6 and
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then x² - 3 is to 18 right and if I was
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to do this now what would that give me
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two and then what
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to the power of three three right three
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but because this exponent is odd what
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will I do i'm going to add absolute
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value absolute value to avoid because I
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want to make sure that the number that
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I'm getting is not negative so this is
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why I'm putting absolute value any
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anytime the exponent is odd you want to
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do that yeah so would you do that for 3
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A4 B5 b5 yes technically I should have
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done it for B here i could have I should
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have put D3 to the five here because But
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you wouldn't do it for the rest of I
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wouldn't do it for I wouldn't do it for
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here because all the exponent are what
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even I mean like you wouldn't do for No
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I would do it for because any any number
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raised to the power of even number is
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always going to be what positive right
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but here good observation I have to put
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in
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B35 so that is an absolute value right
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so this is technically section 6.4 the
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end very simple one so now I'm going to
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give you some work to do so that's it