WEBVTT

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We're asked to multiply
and to simplify.

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And we have x squared minus
the principal square root

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of 6 times x squared plus the
principal square root of 2.

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And so we really just
have two binomials,

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two two-term expressions
that we want to multiply,

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and there's multiple
ways to do this.

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I'll show you the
more intuitive way,

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and then I'll show
you the way it's

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taught in some
algebra classes, which

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might be a little bit
faster, but requires

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a little bit of memorization.

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So I'll show you the
intuitive way first.

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So if you have
anything-- so let's say

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I have a times x
plus y-- we know

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from the distributive
property that this

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is the same thing as ax plus ay.

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And so we can do the
same thing over here.

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If you view a as x squared--
as this whole expression

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over here-- x squared minus
the principal square root of 6,

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and you view x plus y
as this thing over here,

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you can distribute.

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We can distribute all
of this onto-- let

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me do it this way-- distribute
this entire term onto this term

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and onto that term.

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So let's do that.

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So we get x squared minus
the principal square root

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of 6 times this term-- I'll do
it in yellow-- times x squared.

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And then we have plus
this thing again.

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We're just distributing it.

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It's just like they say.

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It's sometimes
not that intuitive

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because this is
a big expression,

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but you can treat it just like
you would treat a variable over

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here.

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You're distributing it over
this expression over here.

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And so then we have x squared
minus the principal square root

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of 6 times the principal
square root of 2.

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And now we can do the
distributive property again,

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but what we'll do is we'll
distribute this x squared

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onto each of these terms and
distribute the square root of 2

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onto each of these terms.

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It's the exact
same thing as here,

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it's just you could imagine
writing it like this.

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x plus y times a is still
going to be ax plus ay.

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And just to see the pattern, how
this is really the same thing

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as this up here,
we're just switching

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the order of the multiplication.

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You can kind of view it as we're
distributing from the right.

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And so if you do this, you
get x squared times x squared,

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which is x to the fourth,
that's that times that, and then

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minus x squared times the
principal square root of 6.

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And then over here you
have square root of 2 times

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x squared, so plus x squared
times the square root of 2.

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And then you have square root
of 2 times the square root of 6.

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And we have a negative
sign out here.

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Now if you take the
square root of 2--

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let me do this on the
side-- square root of 2

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times the square root of 6, we
know from simplifying radicals

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that this is the exact same
thing as the square root of 2

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times 6, or the principal
square root of 12.

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So the square root of 2
times square root of 6,

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we have a negative
sign out here,

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it becomes minus the
square root of 12.

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And let's see if we can
simplify this at all.

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Let's see.

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You have an x to
the fourth term.

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And then here you
have-- well depending

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on how you want to view
it, you could say, look,

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we have to second degree terms.

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We have something
times x squared,

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and we have something
else times x squared.

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So if you want,
you could simplify

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these two terms over here.

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So I have square
root of 2 x squareds

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and then I'm going to subtract
from that square root of 6

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x squareds.

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So you could view this
as square root of 2

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minus the square root of 6, or
the principal square root of 2

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minus the principal square
root of 6, x squared.

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And then, if you want,
square root of 12,

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you might be able
to simplify that.

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12 is the same
thing as 3 times 4.

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So the square root of 12
is equal to square root

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of 3 times square root of 4.

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And the square root of 4, or
the principal square root of 4

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I should say, is 2.

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So the square root of
12 is the same thing

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as 2 square roots of 3.

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So instead of writing the
principal square root of 12,

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we could write minus 2 times
the principal square root of 3.

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And then out here you have
an x to the fourth plus this.

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And you see, if you
distributed this out,

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if you distribute this x
squared, you get this term,

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negative x squared,
square root of 6,

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and if you distribute it onto
this, you'd get that term.

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So you could debate which
of these two is more simple.

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Now I mentioned
that this way I just

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did the distributive
property twice.

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Nothing new, nothing fancy.

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But in some classes, you will
see something called FOIL.

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And I think we've done
this in previous videos.

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FOIL.

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I'm not a big fan of
it because it's really

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a way to memorize a process
as opposed to understanding

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that this is really just from
the common-sense distributive

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property.

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But all this is is
a way to make sure

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that you're multiplying
everything times

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everything when
you're multiplying

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two binomials times
each other like this.

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And FOIL just says, look,
first multiply the first term.

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So x squared times x
squared is x to the fourth.

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Then multiply the outside.

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So then multiply--
I'll do this in green--

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then multiply the outside.

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So the outside terms are x
squared and square root of 2.

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And so x squared times
square root of 2--

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and they are positive-- so
plus square root of 2 times x

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squared.

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And then multiply the inside.

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And you can see
why I don't like it

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that much is because you
really don't know you're doing.

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You're just applying
an algorithm.

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Then you'll multiply the inside.

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And so negative square
root of 6 times x squared.

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And then you multiply
the last terms.

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So negative square root of
6 times square root of 2,

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that is-- and we
already know that-- that

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is negative square root of
12, which you can also then

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simplify to that expression
right over there.

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So it's fine to use
this, although it's good,

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even if you do use this, to
know where FOIL comes from.

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It really just comes from
using the distributive property

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twice.