 ```Radical Functions
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8.3
Objectives
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expressions must have the same index. To multiply two
radical expressions with the same index, multiply the
After multiplying the radicands, simplify the result if
possible.
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Multiply the following and simplify the result.
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Example 1 – Solution
There are no square factors, so it is simplified.
20 has a perfect square factor, so simplify.
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Example 1 – Solution
cont’d
36 and a2 are perfect squares, so simplify.
20 has no perfect cube factors,
but the m4 can be simplified.
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the Denominator
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Dividing Radicals and Rationalizing the Denominator
Division inside a radical can be simplified in the same way
that a fraction would be reduced if it were by itself.
This follows from the powers of quotients rule for
exponents.
You can use this rule to simplify some radical expressions
that have fractions in them.
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Dividing Radicals and Rationalizing the Denominator
Please note that you can simplify only fractions that are
That is, you cannot divide out something that is inside the
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Example 4 – Simplifying radicals with division
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Example 4 – Solution
Reduce the fraction and then
The fraction does not reduce, so separate
the radical and then simplify each
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Example 4 – Solution
Reduce the fraction.
Since the fraction does not reduce further,
Reduce the fraction.
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Dividing Radicals and Rationalizing the Denominator
Clearing any remaining radicals from the denominator of a
fraction is called rationalizing the denominator.
This process uses multiplication on the top and bottom of
the fraction to force any radicals in the denominator to
simplify completely.
The key to rationalizing the denominator of a fraction is to
multiply both the numerator and the denominator of the
fraction by the right radical expression.
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Example 5 – Rationalizing the denominator
Rationalize the denominator and simplify the following
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Example 5(a) – Solution
Multiply the numerator and denominator by
the denominator. This is the same as
multiplying by 1.  