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Let's learn how to rationalize a
denominator. For example, let's
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rationalize the denominator of this
expression and write our answer in
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simplified form. Looking at this
denominator here, we have an irrational
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number here and an irrational number here.
Remember to rationalize the denominator
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means to make the denominator a rational
number. And we can do that by multiplying
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both the numerator and denominator by the
conjugate of the denominator. And
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remember, the conjugate is the same
expression but we just changed the minus
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to a plus. And if we multiplied our
denominator by this, we need to multiply
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our numerator by this as well. Let's put
parenthesis here so we know that we're
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going to foil both the numerator and
denominator. So, what do we get? Looking
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at the numerator here we have three square
root of two times three square root of
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five.
And the square root of two times square
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root of five is square root of ten and
three three is nine.
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So, this is equal to nine square root of
ten.
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The outer term is twelve square root of
four, the inner term is -six square root
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25, and the last term is -eight square
root of ten.
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What about the denominator? When we foil
this out, we have nine square root of 25.
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The outer term is +twelve square root of
ten.
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The inner term is -twelve sqaure root of
ten, and the term is -sixteen square root
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of four.
Now, by multiplying by the conjugate,
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these outer and inner terms in the
denominator will cancel. And what are we
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left with? We have nine square root of ten
and then +twelve square root of four.
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But the square root of four is two, so we
have +24 and then, -six square root of 25
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and square root of 25 is five, so this is
-30, and then -eight square root of ten
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divided by, we have nine times the square
root of 25 or nine times five, which is
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45. And then, -sixteen times square root
of four, but square root of four is two,
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so its -32.
Now, let's combine like terms. This nine
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square root of ten - eight square root of
ten, gives us +one square root ten.
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So, this is equal to square root of ten
and then 24 - 30 is -six and the
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denominator is thirteen.
And we have rationalized our denominator
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and written our answer in simplified form.
So, this would be our answer. And this is
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how we rationalize a denominator. We
multiply both the numerator and
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denominator by the conjugate of the
denominator. Thank you and we'll see you
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next time.