Algebra 1 Common Core Student Edition, Grade 8-9

Conjugates are the sum and difference of the same two terms. For example, V7 + V3

and V7 - V3 are conjugates. The product of conjugates is a difference of squares.

(V7 + V3)( V7 - V3) = (V7)2 - (V3)2

=7-3=4 The p ro d u ct o f th e c o n ju g a te s has no radicals.

You can use conjugates to simplify a quotient whose denominator is a sum or difference

of radicals.

How do you
rationalize the
denominator?
Multiply by the conjugate
o f th e d e n om inator. If
th e d e n o m in a to r has
the form a - b, the
c o n ju g a te is a +b.

Pr o b l em 4 Rat i o n al i zi n g a D en o m i n at o r Usi n g Co n j u g at es

What is the simplified form of^10

V7 - V2'

10 _ io _ V7 + V 2 Multiply the numerator and denominator by the

V 7 - V 2 V 7 - V 2 V7 + V2 conjugate of the denominator.

10( V7 + V2)

7-2

10(V7 + V2)

_ 5

= 2(V7+ V2)

= 2 V7 + 2 V2

Multiply in the denominator.

Simplify the denominator.

D iv id e 10 a n d 5 b y th e c o m m o n fa c to r 5.

Simplify the expression.

Go t I t? 4. What is the simplified form of

Vl0 + V5 '

How do you begin

this problem?

Since th e re cta ng le is

a golden rectangle,

the length divided by

the w id th has to equal

1 + V5

2 '

Golden rectangles appear frequently in nature and art. The ratio of the length to the

width of a golden rectangle is (1 + V 5 ): 2.

Pr o b l em 5 So l v i n g a Pr o p o r t i o n I n v o l v i n g Ra d i c a l s QZ 5 )

Biology Fiddlehead ferns naturally grow in spirals that fit into

golden rectangles. What is the width w of the fern shown?

1 + V 5

m(l + V5) = 8

w =

w =

w =

1 + V 5

8

W r ite a p r o p o rtio n.

Cross Products Property

Divide each side by 1 + V 5.

1 - V 5

w = ■

1 + V 5 1 - V 5

8 - 8 V5

1 - 5

8 - 8 V 5

Multiply the numerator and

denominator by the conjugate

of the denominator.

Multiply.

Simplify the denominator.

w = — 2 + 2 V5 ~ 2.5 Simplify. Use a calculator.

The width of the fern is about 2.5 cm.

628 Chapter 10 Radical Expressions and Equations