The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

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WORKSHEET 7.7: USING CONJUGATES TO SIMPLIFY
RADICAL EXPRESSIONS
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If the denominator of a radical expression contains the sum or difference of a number and a
radical, use a conjugate to simplify the expression. A conjugate is an expression with the
opposite operation of the original expression. For example, the expressions 5 +


3 and
5 −


3 are conjugates. Follow the steps below to use conjugates to simplify radical
expressions:


  1. Identify the conjugate of the expression in the denominator.

  2. Multiply the numerator and denominator by the conjugate.

  3. Simplify the product.


EXAMPLE
Simplify

3

2 +


3

.

2 −


3 is the conjugate of the expression in the denominator.

3
2 +


3

·

2 −


3

2 −


3

3(2−


3)

(2+


3)(2−


3)

=

6 − 3


3

4 − 2


3 + 2


3 −


9

=

6 − 3


3

4 − 3

6 − 3


3

1

= 6 − 3


3

DIRECTIONS: Simplify each expression.






2

4 +


3





5

7 −


5





10

5 +


2





15


7 − 5





4

3


2 − 10





3

4 − 7


3

CHALLENGE:Explain why multiplying by a conjugate eliminates the radical
sign.

265

Copyright


©


2011 by Judith A. Muschla, Gary Robert Muschla, and Erin Muschla. All rights reserved.

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