Chapter 8: Inverses and Radicals

Lesson 6: Quotients with Radicals

Mrs. Parziale

Quotients with Radicals

Given the fraction:

To write an equivalent form with a rational number as the denominator:

• Multiply both the numerator and the denominator by

• Does this change the value of the fraction?

• Find decimal approximations of and the final rationalized equivalent.

• Conclusion:

Rationalizing the denominator of a fraction means: write an ____________________ form of the fraction with a rational number as the denominator.

equivalentExample 1:

3

2

2

2

NO3

2

They are the same value.

Rationalizing the DenominatorRationalizing the Denominator

• Example 2: Rewrite each expression without a radical

sign in the denominator:

To rationalize the denominator of a fraction whose denominator is multiply both the numerator and denominator by (a > 0)

because .

To rationalize the denominator of a fraction whose denominator is multiply both the numerator and denominator by (a > 0)

because .

a

a1

a

a

30

10

Example 2, cont.

1

5b

Conclusion:For all positive numbers x:

Conclusion:For all positive numbers x:

1 x

xx

7

20

25n

Rewrite each expression without a radical sign in the denominator:

Example 3Example 3

• Rationalize the denominator in the fraction:

where x > 03

6

9xSimplify first and then rationalize the denominator.

Example 3, cont.Example 3, cont.

• Rationalize the denominator in the fraction:

where x > 03

6

9xRationalize the denominator first and then simplify.

To simplify a denominator of the form multiply it by its conjugate or To simplify a denominator of the form multiply it by its conjugate or

a ba b a b

Example 4: Rewrite the expression without a radical sign in the denominator:

2

1 5

a)

Example 4, cont.Example 4, cont.

• b)

• c) 2 3 5

2 3 5

4

8 5

ClosureClosure

• Why do we rationalize the denominator?

• Explain how to rationalize the denominator in this example:

• Simplify and rationalize the following expression:

4

2

x

x

3

8

16

x

x