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Rational Exponents & Radicals
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Apr 25, 2025
Section 6.6. In this lecture, we'll learn how to write expressions with rational exponents in radical form and vice versa. We'll also learn how to simplify expressions in exponential or radical form. Chapters:
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we only have we don't have that much
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time left four or five weeks of really
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and there's a lot of little things
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happening here and there rams ram this
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ram that so let's just we're going to
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try to stay focused get our work done so
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we can be done with it right so I'm
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going to try and teach you the the fine
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art of uh algebra 2 i'm going to try to
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teach you like the things that you need
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to know before you leave this class i
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don't want to Uh-huh what's our title
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can you give us like a good review yeah
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I do that every year i give review like
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an it's like it's going to cover like
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what we cover from the mid midterm all
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the way to the end right shut down those
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two computers
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how many questions are going to I don't
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know probably 40 questions 50 50 maybe
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40 questions okay 40 questions or
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something like that all right so today's
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section is going to be about rational
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exponent and we've already talked about
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radical uh numbers right square root
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anything with like fifth root and all of
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that now there's another way to write
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radicals we use this time we use what we
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call it
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exponential form okay so now for example
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if I give you this example here I say
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write x to the 1 / 6 in radical form
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this becomes what the six root of x this
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is how you go from one to the next okay
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and then here I say write the fourth
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root of z in the exponential format it
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becomes z to the 1/4 now there is a
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general formula that you can use to
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figure this out right so which is this
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if I have uh b roo of x to the a
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this is the equivalent of x to the a
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over b example the fifth root of x² is
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basically x to the 2th right so that's
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all you need to know with going from
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radical x radical form to the
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exponential form for example if I give
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you this here
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um and I want you to write this in the
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exponation form what would that be
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x and 5 over 7 right x 5 7 what if I
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have this now
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uh and I want you to go and write this
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in a radical form so what would that
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be b3
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B to 1/3 right so you can go from one to
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the next it's really easy to convert
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them okay and why this is useful you're
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going to see that in a little bit here
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right for example the first subsection
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here the second subsection is how do you
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evaluate an expression with a rational
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exponent right how do you evaluate it
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when you hear the word evaluate that
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means you need to calculate what it is
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we need to find it right we need to find
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what it is this is 81 yeah would that be
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81 or 81 no no no it's not No it's not
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negative 81 what about the negative i'm
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I'm getting there you're jumping ahead
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of time right so I've evaluate right an
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expression with a rational exponent the
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rational exponent here is 1 to the
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4gative 1/4 right so the first thing I'm
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going to do is remember whenever you
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have a negative we need to make sure at
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some point that negative is you're going
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to you're going to get rid of it okay
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but before you do that you can rewrite
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this without a negative exponent so what
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do you do it becomes what you remember
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from last uh before you left on the
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break one over what 81 81 to the 1/4
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right but then we are not done so what
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we're going to do is we're going to try
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to evaluate this okay so I'm going to
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break down 81 if I break down 81 81
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gives me what 9 9 * 9 which is also 9
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square
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9 square it's a good thing but we want
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to go to the we want to make sure when
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we do that we use uh the prime number 3
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to the what 3 to the 4th right so
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81 is 9 * 9 and this is 3 3 and 3 so
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it's 3 to the 4th so that means I can go
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and replace this by 3 to the 4th the
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whole thing to the power of what
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1/4 so you see what's happening here
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right now you remember we talked about a
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power of a power right so what do you do
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with the four and 1/4 you can multiply
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them right we're going to multiply these
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here because when you multiply you're
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going to be able to simplify after
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multiplying okay so that becomes what
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that becomes 1 over 3 and then 4 * 1/4
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is 4 which is basically 1 over 3 right
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so basically 81 raised to the power of 1
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/ 4 is 1/3 so this is the simplified
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version of it so this is how you
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evaluate it okay so 81 to the 1/4 became
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1 over 3 you got to go through a
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transformation process recognize that
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this can be broken down and then once
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you break it down you can get to this
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point and then go a little further down
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and then simplify that okay
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example let me give you an example here
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that you guys can work on to figure this
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out so what if I have
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um
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-3125 raised to the power 15
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okay actually let's let me not put a
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negative there so I got
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3,1 125 raised to the power of 1 over 5
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so how would you do this here
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okay
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uhhuh one over now I know you can put
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this in your calculator it's going to
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spit it out but part of this exercise is
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for homework you may not show me your
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work don't put in your calculator and
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just put one like don't do that cuz that
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I'm going to require to see your work
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right so 3125 so if you break down 30
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3125 so how can we break down 3125 why
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goes into five and what uh I don't know
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how 5 and 5 and 6 6 25 right and 625 is
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25 and 25 right uh and now it's five and
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five and this is five and five right how
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many fives do we have 1 2 3 4 5 right so
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we have now this becomes 1 over
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3125 to 15 right and then we can replace
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this by 5 through
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Okay it's 1 over 5 to 5 raised to 1/5
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and then we get to the same result okay
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five multip power so the final result is
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15 now the calculator can do that in a
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split second right this is the power now
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of AI and all these tools we can put
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stuff in program yeah I mean it's still
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artificial intelligence right you put in
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the calculator it's still it's a
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calculator right you put in the it's
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still doing the work for you the
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computer is still doing it so it's still
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artificial intelligence that only counts
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if itself train itself to do the task
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we've programmed every possible outcome
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the same way with the AI too we trained
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him you trained it it did the training
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on it i'm not He's not artificial i
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trained him to do something
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he's not artificial i don't think so i
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don't know i think he might be
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artificial matt might
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All right so now next thing we're going
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to try to go for the next one 216 to the
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2/3 as you can see there this is a
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little different right so I'm going to
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choose the same route i'm going to break
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down 216 right so if you break 216 with
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2/3 so how would you break down 216
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uh okay so what goes into four and what
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four and um
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four and 54 54 is that what you got
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okay
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and then 54 is what six and nine right
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and 9 is three and three and six is
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three and two right and four is what two
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and two now watch this how many twos do
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I have three how many threes do I have
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so we're going to rewrite this as right
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two to the what third
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time 3 to the 3r the whole thing raised
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to the power what 2/3 right so now what
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we're going to do is each one is going
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to be multiplied by the exponent we're
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going to do them one at a time okay so 2
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to the 3 * 2/3 is going to give you
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what 23 to the 6 over three right cuz
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this is this and then the same thing
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here three this and this gives you three
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to the 63rd right that's your answer No
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we're not done 6 over 3 is what two so
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it's 2 to the 2 * 3 to the 2 which is
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basically 4 * 9 that is what is 4 * 9 36
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right 4 * 9 36 again yeah
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right here 6 over 3
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right 6 over 3 is 2 6 over 3 is 2 so
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it's 2 * 3 square which is 4 * 9 that is
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36 all right so basically we are just
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using that transformation to get to the
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result right you got to be able to like
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simplify numbers write numbers using the
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what what do you call that again this is
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the prime factorization you need to be
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able to do this okay prime factors prime
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factors we did that in the pre-ex i'm
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doing it now right all right so now the
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next thing we're going to learn how to
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do is we're going to learn how to
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um simplify expressions with rational
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exponent now the first one is pretty
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straightforward right i have a to the
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27th time a 47 since the base is the
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same what do I do um you can add the I
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can add the exponent right they have the
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same base so this becomes what a 27 6
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over 7 plus 47 right because the base is
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the same you just add the exponent so
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that becomes 8 to the 6 over 7 right and
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I'm done right i'm done with this one
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now the next one is the one that can is
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going to cause us to do a little bit of
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a little bit more work cuz again we're
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going to have to use uh the conjugate
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expression here right now as you can see
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what kind of exponent is this negative
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negative i have a negative exponent so I
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need to f my first transformation
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fraction is to get this distance turns
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into a fraction right so that's going to
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give me one over what b to 5 6 right but
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now I want to rationalize the
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denominator right if you went to is
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called the denominator right so I want
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to rationalize
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the denominator right so what do I do i
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when I say rationalize that means what I
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do not want to have an exponent that is
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this is not a r this is I want to
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rationalize it I want to get turn this
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into a real number right so this is b to
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the 56 so how much do I need to add to
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these to change that one one so this is
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what I'm going to add what b to
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the six and b to the one six right
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because now what's going to happen is
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this this plus this if I add the
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exponent I'm just going to get what b^ 6
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over 6 which is B to the 1 right again
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I'm trying to rationalize the
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denominator so on the top I still have
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my B to the 1 / 6 and on the bottom if I
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do this here I'm going to add the
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exponent i get B to the 6 / 6 so this is
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basically B to the 1 / 6 over B and this
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is where you start because you
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rationalize the denominator okay if you
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ask me I think it's a very silly way of
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doing things because what's the point of
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I mean I'm good I'm good with just
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keeping a negative exponent but it's
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just standard say we have to transform
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it because to me this looks more complex
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than that right why would I want to do
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this when I just stop here and know that
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this is my expression but that's what it
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do so we have to do it okay now now
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we're going to learn how to simplify
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radical expressions we've done this in
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our last session but we can still do it
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again right now if you notice
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here right can I divide 27 three
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no why why can I not do that
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they're different roots right this is
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what the fourth root and this is just
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what square root right so I cannot
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divide them but if this was just
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roo of 27 over 3 I could have just
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done it and that would have given me
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what square root of
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nine which is three but unfortunately we
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have this here so we can't do that right
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so you have to make sure you can't
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divide these two if they do not have the
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same roots you can't do that you cannot
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i repeat that again you can't right so
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now what we do is this we're going to
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use the um exponential of form to do
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this right so how do I write 27 uh 4 27
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what 27 to the power of what
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1/4 right and then this is just going to
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be what square root of 3 is 3 to what 12
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3 to 1/2 right now I can work with this
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now I can I can change 27 again i'm
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going to use the same kind of uh
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process okay so I'm going to go here and
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I know that
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27 is 3 and 9 and is 3 and 3 so it's 3
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to the 3 right so I'm going to rewrite
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this as 3 to the 3r the whole the whole
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thing to the 1/4 over 3 to the 1/2 now
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we're not done yet i can still transform
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this here right so this
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is 3 to the 34 and why is that so
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because I multiply this right power of
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power you have to multiply them over 3
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to the 12 now what do I do here i have 3
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to the 3 4 over 3 to the 1/2 how do I
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simplify this
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you have to change the 12 to the
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denominator for 4 we can do that or we
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can just subtract them right
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well you can do that too because this is
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3/4 and this is 1/2 so I can find the
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common denominator right what's the
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common denominator here how do I change
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one two4s two four so that becomes 3 to
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the 34 right over 3 to the 24 and now I
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can subtract it right this minus that
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that gives me what 14 3 to the 1/4 right
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34 - 24 that's 3 to the 1/4 and I'm I'm
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done i can stop here
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okay so again you have to recognize we
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using multiple techniques here to solve
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this you have to be able to understand
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that you need to turn this into a a
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rational um exponential um form and then
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second you need to recognize that you
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need to break down the 27 into it prime
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numbers and then once you divide we've
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learned the division rule of exponent by
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subtracting them okay and make sure you
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never get a negative exponent because
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technically they say that this is not
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how you write things okay now what about
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this one here i have cubic root of 64 Z
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to the 6 so what do I do here
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i'm going to use Yeah
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break down the 64 right we're going to
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start by breaking down the 64 right can
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I erase this yes so we're going to start
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by breaking down the
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64 okay so 64 oh you know 64 is 8 and 8
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right and I know 8 is 2 to the 3 because
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it's 2 * 2 * 2 right three twos and this
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is the same here two two and two so 84
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64 is basically two raised to the power
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of six okay so now I can work with this
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right is that Z or two that's Z so that
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becomes now 64 i put this things in here
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like this right 6 to the^ of
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what 1 over three everybody agrees with
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me on that right because if I put this
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in bracket I can raise the whole thing
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to the power of 1/3 because the outside
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is three okay I can do that if I want
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now I know
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that 64 is 2 to the^ six okay so I'm
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going to change this into 2 to the
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6 Z to the 6 the whole thing to the 1/3
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and then my last step will be to just
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multiply the exponent right so now it's
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going to give me 2 to the 6 over 3 times
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Z to the 6 over 3 and that just give me
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2^ 2 right because 6 over 3 is 2* Z
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^2 and then 2^2 is 4 so my final result
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is 4 Z squ all right does that make
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sense now when it comes to this section
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the way you going to get better do more
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work can't get better by just focus on
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my examples these are just too little
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you have to be able to like do your own
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stuff okay so now I'm going to give you
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some things to work on here real
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quick and then uh uh let's see here what
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I want oh there's one more thing that I
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want to work
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on uh maybe I will save that for
18:28
tomorrow i will save that for tomorrow
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wait a minute goodness gracious so quick
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