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How To Simplify, Add, Subtract, Multiply Radical expressions & Use The Conjugate Expression
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Apr 9, 2025
In part2 of this lecture , we'll learn how to simplify, add, subtract and multiply radical expressions and we'll also talk about the use of the conjugate expression to rationalize the denominator of radical expressions.
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0:00
he didn't take my math test
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Who jesus Yeah he doesn't need to He
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already knew He already knows what to do
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with it
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What do you think he would get do you
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think he would just get 100 i think he'd
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be Jesus for all the answers Do you
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think he would be one of those people
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that doesn't need to study he just knows
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Yeah because he No that is not true He
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needed because he was a man and in his
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human nature he was limited in his he
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was limited Do you think do you think he
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was truly human if Jesus took a math
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test wasn't he half human half did he go
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half human that is called a monopite
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heresy He's not half and half Wait wait
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Did he go to high school
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he went to Jesus Come on now We were at
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high school in this time They
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had if Jesus took a bad test do you
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think he would just get 100 cuz All
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right You got you got you got you got a
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perfect man Of course he going to get
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he's going to ace all his test That's
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why he put everybody to shame because he
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was he he
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studied He wasn't just sitting around be
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like "Okay no." Cuz he was limited in
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his he was limited as a human He was
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truly human right because when it was
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baby Jesus had to be helped cuz he was a
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baby Yeah So he wasn't like a baby that
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could do like Superman No he was a baby
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So
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right You know that he got tired right
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jesus got tired because he was a human
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And just like a human he had human like
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um everything that we went we went
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through as humans he went through it
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because he was a human in his nature
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That's what that's why he understands
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the human experience because he wasn't
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just like a superhuman He had a full
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human nature and a full divine nature
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That's called that's called he still had
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to study right but you talk about what's
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called a monopite heresy
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We're going to have to stone you now
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Unfortunately
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it's a heresy She's doing heresy We got
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a heresy And then the guy that the guy
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that came up with the hair was called So
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look
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at the blood Waiter Wait Half human and
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half god is like that is not that's like
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Moana Yeah That's that's that's heresy
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That's Hawaii You know what a heresy is
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right uh no Heresy When you hear the
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wordes against God right it's something
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that is heresy Yeah Orthodox Third
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commandment Thou shalt not lose God No
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it's not third Orthodoxy right that is
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the when you hear orthodoxy this is
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right thinking on video
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but that's fine it's just among the 10
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it don't matter anyway that so heresy is
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like a lie something that is not true is
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false false teaching and then orthodoxy
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is right teaching right now back to
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radicals right we have root of 98 minus
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2 root of 32 Right the way we solve this
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is this We got to make sure the number
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that is under the radical are identical
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If they're not identical we cannot do
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this Right so that mean we have to break
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down the radical We have to break them
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down right so I'm going to break down 98
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If I break down
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98 98 gives me what i mean I already
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know since eight is divisible by what
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two right two is what 49 I know that 49
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is a perfect first square perfect square
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So I'm going to go back here I'm going
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to go 49 * 2 right and here I'm going to
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do the same thing with 32 I'm going to
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break down 32 into something that has a
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perfect square which is what 16 * 32id
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two right 16 So I can deal with this I
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know how to work with this So now what's
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of 49
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seven Seven So that's out That's going
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to go out So I get seven square root of
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two And what's of 16 four That's going
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to be 2 *
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4 of two Right now I have seven
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square of two -
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8 2
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So now if you look at it the outer
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numbers are the only one that I'm going
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to deal with right because they have the
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same number under the radical So the
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only thing that you're subtracting is
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the outer not the inside number You do
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have to make sure that the inside
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matches For example you can't
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72us - 8 That's impossible right the
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number that's under the radical has to
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be the same Now all we have to do now is
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subtract 7 - 8 And that gives you what
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or if you want negative two whichever
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one either whichever way you want to put
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it right so it's either 1 square root of
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two or negative root of two All right so
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this is how you subtract or add radicals
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Now I'm going to give you another
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example Yes What are you doing over
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i don't know I ain't got nothing to do
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Probably going to hide some eggs
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I don't know I could beat pickle ball We
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should do a a staff or student pickle
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ball Nobody know that I'm going to be
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smoking everybody's face No one beat me
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on that Be humble Mr
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No humility on that Let's do a humidity
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on that I'll get
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everybody before the fall Oh listen
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Anybody get anybody's going to get
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smoked on that one what
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i feel like I would be in your I feel
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like
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So let's try to work on this one here
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right let's try to work on this one So I
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have How many do I have now three I have
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three radicals right i have three
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radicals here And I'm I'm gonna make
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sure I'm matching them I want to make
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sure the inner the number that's inside
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the radical is the same Right so I'm
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going to break down 12 into a perfect
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square times a number Right it will be
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four * what four * three And 27 will be
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9 * 3 Right as you can see my inner
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number I want it to be like something
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with a three because this this all I
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want multip can be multiplied by three
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Right and 128 128 Again if you add the
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digits right 8 + 2 is 10 10 + 1 is 11
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This number is not divisible by three So
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I'm going to just do two times uh
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two is 64 right and 64 is just 8 and 8
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right so I'm not going to waste my time
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So if you look at it now I'm going to
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have 5 4 * 3 right + 2 9 * 3 -
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8
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* uh - 64 * 2 which is also a perfect
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square I know 64 is a perfect square
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right so if I take them out one at a
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time right this is going to go out it
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gives me what 5 * what two square root
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of three This goes out It gives me two *
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3
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And here this goes out It's just 8
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square root of two Right now if you look
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correctly right five and two that is 10
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square root of three 2 and 3 that is
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6 3 - 8 2 right now if you look
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here I have to make sure the number that
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is inside the radical matches right that
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that now here there's nothing I can do
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here because it's a square root of two
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so the other one that I can combine is
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this one this is just like working with
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like terms right so that will be
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16 3 -
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8 so that's basically basically how you
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add and subtract numbers with radicals
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Right now multiplication Now how do you
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multiply numbers with radicals right
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we're going to do that too
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Now we've done it The one that we've
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done it's like it's a little different
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This one is going to be
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involving right so we're going to write
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out to multiply
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multiply radicals right
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now watch what happens here right cool
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That's enough
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So 4 3
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right plus
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5 and then times
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um
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3 - 6 Now this involves a little bit of
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we're going to be doing using the
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distributive law here right so I'm going
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to go step by step So it's going to be
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this times this right
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this times this here So you see again
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this is why it's good to learn how to do
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So it's going to be
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4 right times
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3 That's just this And then now I have
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minus minus this times that right minus
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6 times 4 3 That's just the first
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portion of it Right now I'm going to
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have to do the same thing here This and
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this So it's going to be plus
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5*
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3 and then lastly - 6 *
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5 2 So it's a distributive law It
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is extended Right so a lot of things
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that we have to worry about here Now if
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I look at it now I'm going to have to do
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them singly like every one of them
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separately Right so I have four here
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Four square root of 3 * 3 I'm going to
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multiply the outer numbers and the inner
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numbers Right 4 * 3 is what that's 12
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square root of 3 *
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2 six Right so it's 4 * 3 and then the
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you multiply them together So this is 12
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square root of 6 Now here I only have 6
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* 4 that gives me what
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24 of three right because I'm seven this
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times this Now here I have something
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that is going on that's going to be
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interesting here Right so I have four 15
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square root of 4 right cuz of two *
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square of two is square root of 4 And
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then lastly -
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30 of two Now I have to see if there's
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anything else to do here that I can do
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to simplify this Right i already know
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what 4 is That's 2 right so this is 12
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6 - 24 3 + 15 * 2 - 30 2
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and I can do this one Oh that's how you
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do it Times two
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Mhm Now this is just 30 minus 30 2
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And then this is where you stop because
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there's nothing else you can combine
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here right this is square root of two
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This is square root of three This is
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square root of six There's no breakdown
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that's going to happen here So this is
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my final result right there's nothing
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else to do here You can stop here Okay
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And then
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lastly we've already done this before
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but we still going to do it again We're
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going to learn how to find the conjugate
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expression which we did before right we
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did it with
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uh uh we open the window the kids So
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we're going to learn how to welcome my
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little expression right anybody else
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rank this so be good
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All right
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Is he winning
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all right So now we have two right it's
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the last thing that we going to learn
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today And then we want to call we want
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to rationalize the denominator We did
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that before in our in our previous
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section How do you rationalize it right
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you want to
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rationalize the
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denominator right you want to
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rationalize the denominator So basically
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what you want to do is this You want to
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find a real number that's going to
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replace this expression here with a
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radical So to rationalize this it's a
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very easy process We're going to
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multiply this by what we call the
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conjugate expression It's called the
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conjugate conjugate expression right
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conjugate expression And what is it is
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is probably is just the the pretty much
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the the conjugate expression So you
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going to multiply this by 5 + 1 and
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square root of 5 + 1 It's pretty much
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the opposite You just switch this up
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Yeah we've done this Um we've done this
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plus one necessary It's not under the
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square root No the plus one is not under
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the square root It's outside If it's
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under it's a totally different
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expression It's totally it's not 100
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right so why do we need to get rid of it
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why do we need to cuz we can't have with
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square root You can't have a square root
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in the denominator It's just a rule What
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about the negative one which one is Well
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it's not a square root So why would we
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yeah because it's involved in the
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denominator It's not you can't just take
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one You have to rationalize the whole
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thing So it's not going to do five Right
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So now I'm going to multiply that Right
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so it's going to be two square roots of
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five Right i'm using again the root the
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distributive law
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+ 2 Now over 5 * square of 5 We're going
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to do the same thing here that we did
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like this Right so be square of 25
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Square of 5 * 1 is + 1 square of 5
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Square root of 1 * square of 5 is - 1 5
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and then 1 * 1 is 1 You'll notice a
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pattern here What if you do this long a
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lot right so now what's going to happen
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is these two are going to always cancel
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out The middle one are always going to
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cancel out It's just something that's
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going to happen every time when you find
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a conjugate expression right and square
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root of 25 is what five This is 5 minus
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one So the whole thing is 2 5 + 2
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over 4 We are grow we we now know how to
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simplify expression So you don't leave
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this like this cuz two goes into each
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one of them Right so we simplify this by
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two So you get square of five + 1 / two
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And this is called this is pretty much
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we rationalize the denominator Right so
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this is the last thing that we're going
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to learn today And then now I'm going to
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give you what you love to do is
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