WEBVTT
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Hi, my name is Anthony Varela.
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And in this tutorial, I'm going
to talk about rationalizing
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the denominator.
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So first, we're
going to take a look
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at irrational denominators,
what those look like,
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and why we don't like
them in mathematics.
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Then we're going to
talk about conjugates,
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and how we can use
what's called conjugate
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to rationalize the denominator.
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And then we're going to bring
everything together and go
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through an example where we
rationalize a denominator.
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So let's start off by talking
about irrational denominators.
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What do I mean?
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Well, here's an
example of a fraction
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that has an irrational
number in the denominator.
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And in mathematics, we consider
this to be unsimplified.
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And this is actually
because back when
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calculators weren't around, you
can imagine how cumbersome it
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was or hard to divide a
number by an irrational number
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because it has an ongoing,
never-ending decimal pattern,
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which is just, to
be honest, a pain.
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So what we like to do then
is simplify this fraction
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and rewrite it so
that's equivalent,
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but the denominator is a
rational number instead.
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So thinking about how I can
go from the square root of 2,
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which is irrational, to
something that is rational,
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I might multiply the
square root of 2 by itself.
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That will give me the integer
2, which is a rational number.
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But I am writing an
equivalent fraction,
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so I have to multiply both the
denominator and the numerator
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by the same value.
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So you notice here that we're
multiplying this fraction
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by a quantity that's equal to 1.
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It might not look
like 1, but it has
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the same quantity in the
numerator and denominator.
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So this equals 1.
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So multiplying by 1
doesn't change the value,
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but it is going to
change how this looks.
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So now I can multiply
across my numerators.
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And I have an
equivalent fraction here
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that reads 3 times the
square root of 2 over 2.
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It has the same exact value--
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just that we have a rational
number in our denominator.
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So now I'm going to talk
about conjugates, which helps
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us rationalize denominators.
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But first, we have to
understand what a conjugate is.
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So here's an expression, the
square root of six plus 5.
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And the conjugate
of this expression
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is the square root of 6 minus 5.
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Let's take a look
at another example.
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The square root of 3 minus 7.
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And the square root of
3 plus 7 are conjugates.
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So do you think that you can
draft up your own definition
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of what a conjugate might be?
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So a conjugate of a binomial--
that just means something has
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two terms, one term
and another term--
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the conjugate of a binomial is a
binomial with the opposite sign
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between terms.
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So these look nearly identical.
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The only difference is we have
gone from a plus to a minus.
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And in here, we have gone
from a minus to a plus.
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So thinking about
my example before,
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we had just one term,
the square root of 2.
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And this can be considered
its own conjugate.
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So the square root of 2
and the square root of 2
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are conjugates of each other.
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One way to think
about this would
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be the square root of 2 plus
0 and the square root of 2
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minus 0.
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So how do conjugates help
us rationalize denominators?
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So we're going to go
through this example
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here-- the square
root of 2 minus 2
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over the square
root of 6 minus 2.
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And for now, I'd like to
focus on just our denominator.
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And we're going to be using the
conjugate of the square root
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of 6 minus 2 to rationalize
this expression, which
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is the denominator
of this fraction.
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So first, what is the
conjugate of the square root
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of 6 minus 2?
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Well, that would be the
square root of 6 plus 2.
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And we're going to be
multiplying these two
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expressions together.
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And you notice we can
do this using FOIL.
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So recall that with FOIL, we
are multiplying the first two
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terms, then the outside terms,
then the inside terms, and then
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the last terms here.
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So multiplying our
first terms, this
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would be the square root of
6 times the square root of 6.
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That just gives
us the integer 6.
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Well, next, we're going to
multiply the outside terms.
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That would be the square
root of 6 times 2.
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So that's plus 2 times
the square root of 6.
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Now I'm going to multiply
my inside terms, which
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is negative 2 times
the square root of 6.
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So I'll show subtracting 2
times the square root of 6.
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Then multiplying
my last two terms,
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I have negative 2 times positive
2, which is a negative 4.
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Well, notice that
I have a positive 2
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times the square of 6
and a negative 2 times
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square root of 6.
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Those are like terms that I can
combine, but they combine to 0.
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So I really have just 6 minus 4.
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And 6 minus 4 is 2.
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So you see that multiplying
this square root of 6 minus 2
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by its conjugate gives
me an expression that
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has no radical at all.
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So it rationalizes
that expression.
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And just as a reminder, if
you had just a single radical
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like our very first example,
the square root of 2,
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you can multiply
that by a conjugate,
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which it, itself,
the square root of 2.
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And you also no
longer have a radical.
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So multiplying radical conjugate
rationalizes the expression.
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And that's the whole goal
here with rationalizing
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the denominator.
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So let's return to
our original example,
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and we're going to go
through the full process
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of rationalizing
the denominator.
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So I know that I have to
multiply this expression
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by a quantity equal to 1
to keep its value the same.
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And I know I want to use the
conjugate of that denominator.
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So I need to have square root
of 6 plus 2 in my denominator
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and in my numerator.
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So we're going to be
multiplying these two fractions,
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and we're going to be
using FOIL to do this.
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So let's start with going
through our denominator.
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We've already gone through it,
but we'll see the steps again.
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Multiplying the first two terms,
multiplying the outside terms,
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multiplying the inside
terms, and then multiplying
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the last terms.
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Now we're going to do
the same thing FOIL,
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but with our two numerators.
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So multiplying the first
two terms in our numerator,
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we get the square root of 12.
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Multiplying our outside
terms in our numerators,
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we get plus 2 times
the square root of 2.
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Multiplying those two inside
terms, we get negative 2 times
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the square root of 6.
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And then multiplying the
last terms, negative 2
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and positive 2
gives us negative 4.
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So this looks very messy,
but all we want to do now
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is consider our numerator
and our denominator
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separately, combine
any like terms,
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and do some simplification.
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Well, we already
know from our example
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before, that our
denominator simplifies to 2.
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So I'm kind of
skipping this step.
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But you can, of course,
rewind to see how I got that.
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So now let's go ahead and
simplify our numerator.
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And I'm going to start with--
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I see we have an integer there.
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I'm going to start with
that right away, negative 4.
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And I'm actually going to skip
the square root of 12 for now.
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We'll talk about that in a bit.
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So I'm going to write
down my plus 2 times
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the square root of 2 minus
2 times of square root of 6.
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And now this is how I'd like
to write the square root of 12.
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I'm writing that as 2
times the square root of 3.
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And if you're wondering how I
did that, I took the number 12,
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and I broke it down
into 4 times 3.
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So the square root of 12
equals the square root
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of 4 times the square root of 3.
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The square root of 4 is 2.
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So that's how I got this
expression right here.
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So we notice that we've
achieved our goal.
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Our denominator has
been rationalized.
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But we can actually
go one step further.
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And it's not always the
case, but in this example,
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we can simplify more.
00:08:27.900 --> 00:08:30.120 align:middle line:90%
So here's our expression so far.
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And I notice that every
term in my numerator
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has a factor of 2 in it.
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So I'm going to
factor that out--
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so factoring out the
common factor of 2.
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So here's my factor of 2.
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And in parentheses, I've
divided everything by 2 here.
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And now I notice that there's
a common factor in my numerator
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and my denominator.
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Those cancel.
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So now I'm left with an
expression that doesn't even
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look like a fraction at
all, which is kind of cool.
00:08:59.530 --> 00:09:01.710 align:middle line:84%
But we can think
about our denominator
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as being one, which I
don't always have to write.
00:09:04.320 --> 00:09:08.650 align:middle line:84%
So we've rationalized
our denominator.
00:09:08.650 --> 00:09:11.100 align:middle line:84%
So let's review
today's tutorial.
00:09:11.100 --> 00:09:15.450 align:middle line:84%
The conjugate of a binomial is
a binomial with opposite signs
00:09:15.450 --> 00:09:17.070 align:middle line:90%
between terms.
00:09:17.070 --> 00:09:22.650 align:middle line:84%
So a plus b and a
minus b are conjugates.
00:09:22.650 --> 00:09:25.590 align:middle line:84%
We used FOIL to
multiply two binomials.
00:09:25.590 --> 00:09:29.670 align:middle line:84%
The process is multiplying the
first terms, the outside terms,
00:09:29.670 --> 00:09:33.750 align:middle line:84%
the inside terms, and
then the last terms.
00:09:33.750 --> 00:09:37.230 align:middle line:84%
And remember, that
multiplying radical conjugates
00:09:37.230 --> 00:09:39.270 align:middle line:84%
rationalizes the
expression, which
00:09:39.270 --> 00:09:43.170 align:middle line:84%
is the goal of rationalizing
the denominator.
00:09:43.170 --> 00:09:44.760 align:middle line:84%
So thanks for
watching this tutorial
00:09:44.760 --> 00:09:46.830 align:middle line:84%
on rationalizing
the denominator.
00:09:46.830 --> 00:09:49.220 align:middle line:90%
Hope to see you next time.