Teaching Notes 7.7: Using Conjugates to Simplify
Radical Expressions
Using conjugates to eliminate radicals in the denominator of a rational expression requires
students to find the conjugate and multiply expressions that contain one or more terms. Errors
are made when students do not eliminate the radical in the denominator or when they make
mistakes in multiplication.- Explain that the denominator of a fraction may contain the sum or difference of a number
and radical. If this is the case, students will need to use conjugates to eliminate the radical
or radicals in the denominator. - Explain that students may eliminate a termin binomial multiplication if one operation is
the opposite of the other. Offer this example: (x+3)(x−3)=x^2 − 3 x+ 3 x− 9 =x^2 −9.
Explain that the same principle applies to binomial multiplication with opposite operations
with radicals. Provide this example:
(x−√
3)(x+√
3)=x^2 −x√
3 +x√
3 −
√
9 =x^2 −x√
3 +x√
3 − 3 =x^2 − 3- Explain that expressions such asx−
√
3andx+√
3 are called ‘‘conjugates.’’ Once conjugates
are multiplied, the product contains no radicals.- Review the information and example on the worksheet with your students. Note the use
of the distributive property to multiply the numerators. Remind students that a simplified
expression does not have a radical in the denominator.
EXTRA HELP:
Conjugates are used to simplify radical expressions only when the denominator is the sum or
difference of a radical expression.ANSWER KEY:
(1)
8 − 2
√
3
13
(2)
35 + 5
√
5
44
(3)
50 − 10
√
2
23
(4)
− 5
√
7 − 25
6
(5)
− 6
√
2 + 20
41
(6)
− 12 − 21
√
3
131
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(Challenge)Because conjugates are binomials that differ only by the sign of one term, a radical
and its opposite are added. The sum of the radicals is equal to 0.
------------------------------------------------------------------------------------------264 THE ALGEBRA TEACHER’S GUIDE