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One of the “rules” for simplifying
radicals is that you should
never leave a radical in the
denominator of a fraction.
The reason for this rule is
unclear (it appears to be
a holdover from the days of
slide rules), but it is nevertheless
a rule that you will be expected
to know in future math classes.
The way to get rid of a square
root is to multiply it by
itself, which of course will
give you whatever it was the
square root of. To keep things
legal, you must do to the
numerator whatever you do
to the denominator, and so
we have the rule:
·
Multiply the
numerator and denominator
by the denominator
Example:
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Note: If you
are dealing with an nth
root instead of a square root,
then you need n factors
of that root in order to make
it go away. For instance,
if it is a cube root (n
= 3), then you need to multiply
by two more factors of that
root to give a total of three
factors.
If the denominator
contains a square root plus
some other terms, a special
trick does the job. It makes
use of the difference of two
squares formula:
(a + b)(a
– b) = a2
– b2
Suppose that
your denominator looked like
a + b, where
b was a square root
and a represents all
the other terms. If you multiply
it by a – b,
then you will end up with
the square of your square
root, which means no more
square roots. It is called
the conjugate when
you replace the plus with
a minus (or vice-versa). An
example would help.
Example:
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Given: |
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Multiply
numerator and denominator
by the conjugate of
the denominator: |
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Multiply
out: |
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