Radical Functions
Copyright © Cengage Learning. All rights reserved.
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8.3
Multiplying and Dividing Radicals
Copyright © Cengage Learning. All rights reserved.
Objectives

Multiply radical expressions.

Divide radical expressions.
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Multiplying Radicals
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Multiplying Radicals
To use the product property of radicals, the two radical
expressions must have the same index. To multiply two
radical expressions with the same index, multiply the
radicands (insides) together.
After multiplying the radicands, simplify the result if
possible.
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Example 1 – Multiplying radicals
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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Dividing Radicals and Rationalizing
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
either both inside a radical or both outside the radical.
That is, you cannot divide out something that is inside the
radical with something that is outside of a radical.
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Example 4 – Simplifying radicals with division
Simplify the following radicals.
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Example 4 – Solution
Reduce the fraction and then
simplify the remaining radical.
The fraction does not reduce, so separate
the radical and then simplify each
remaining radical.
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Example 4 – Solution
Reduce the fraction.
Since the fraction does not reduce further,
separate the radical and simplify.
Reduce the fraction.
Simplify the radical.
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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
radical expressions.
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Example 5(a) – Solution
Separate into two radicals.
Multiply the numerator and denominator by
the denominator. This is the same as
multiplying by 1.
Simplify the radicals.
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Example 5(b) – Solution
Multiply the numerator and denominator by
the denominator.
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