Page 708 - Algebra 1

The conjugate of an irrational number in the form a + √ b is a - √ b. The conjugate is used to rationalize the denominator of a fraction when the denominator is a binomial with at least one term containing a radical.
Using Conjugates to Rationalize the Denominator
Simplify. a. _3
4 + √ 5 SOLUTION
Find the conjugate of the denominator. Use the conjugate to write a factor e_quivalent to 1. Multiply the fraction by the factor.
Example
4
Math Reasoning
Analyze Why must conjugates be used when rationalizing denominators with radicals containing binomials?
3
4 + √ 5
3 __
· (4- √5 ) 4 + √ 5 ( 4 - √ 5 )
Theconjugateof4+ √5 is4- √5 .
Use the Distributive Property and the FOIL method to multiply numerators and denominators.
Combine like terms and simplify.
Write the solution as two fractions with the same denominator.
The conjugate of √3 + 1 is √3 - 1.
Use the Distributive Property and the FOIL method to multiply numerators and denominators.
Combine like terms and simplify.
Factor the numerator. Divide. Simplify.
= 12 - 3√5 __
16 - 4√5 + 4√5 - 5 = _12 - 3√5
=
11
- 3√5 _12 _
11
b. _2
11
√ 3 + 1 S_OLUTION
2
_√3 +1 _
=
· ( √3 - 1) √ 3 + 1 ( √ 3 - 1 )
2 __
2√3 - 2
3 - √3 + √3 - 1
_2 √ 3 - 2 = _2
= 2(√3 - 1) 2
= √3 - 1
Lesson 103 693