Chapter 9 Section 4

Dividing Square Roots

Learning Objective1. Understand what it means for a square root to be simplified

2. Use the Quotient Rule to simplify square roots

3. Rationalize denominators

4. Rationalize a denominator that contains a binomial

Key Vocabulary

• simplified square roots

• quotient rule for square roots

• rationalizing a denominator

• conjugate of a binominal

Simplifying Square RootsSimplified Square Roots have

1. No radicand has a factor that is a perfect square

2. No radicand contains a fraction

3. No denominator contains a square root

8 4 2 2 2

12

12

Rule # 3 – Quotient Rule for Square Roots

When square roots contain a fraction divide out the common factors, then use the quotient rule to simplify

Example:

Quotient Rule for Square Roots

0 0a a provided a and bbb

40 408

5

81

5

Example:

Example:

Simplify Square Roots Using Quotient Rule

25 25 581 981

6432

2m n1

21n

2

1

2 2

321

32 16 2 4 2

m

m m m

Example:

Example:

306

5x2x

y1

8z

5

z

3

1x 7

1y

6

7

y

z

2 2

6 36

1

6 6 6x z x z x zy yy

3 375

1

7525

1 125 5

Simplify Square Roots Using Quotient Rule

Example:

4816

9x3

2x

y1

3 6

1x 8

1y

6

3 2

6 36

16 16 4

y

x x x x xy yy

Simplify Square Roots Using Quotient Rule

Rationalizing the denominator means to remove all radicals from the denominator.

To rationalize a denominator with a square root: Multiply both the numerator and the denominator of the fraction by the square root that appears in denominator or by the square root of a number that makes the denominator a perfect square

Example:

Rationalize Denominators

1 1 6 6 666 6 6 36

Example:

Example:

Rationalize Denominators

3 3 7 21 217 77 7 49

2 2 2 2 2 232 832 32 2 64x x x x

Conjugate is a binomial having the same two terms with the sign of the second term changed.

Binomial Conjugate

Multiply both the numerator and denominator by the conjugate of the denominator.

When multiplied using FOIL the sum of the Outer and Inner terms is 0

Rationalize Denominators that contain Binomials

4 2 4 2

Example:

Example:

8 8 3 5 24 8 53 5 3 5 3 5 9 25

24 8 5 249 5

6

8 54

1

6 8 5

10 2 710 10 2 72 7 2 7 2 7 4 49

10 2 7 10

2 7

2

2 7

5

1

2 2 7

Rationalize Denominators that contain Binomials

Example:

Example:

5 5 4 15 4 5 5 154 15 4 15 4 15 16 30

4 5 5 15 4 5 75 4 5 75 4 5 3 25 4 5 5 316 15 1

2 2

2 2 2

a a ba a a ba b a b a b a b

a a b a a b a a aba b a b a b

Rationalize Denominators that contain Binomials

Remember• Rule # 1 - Product Rule for Square Roots

• Rule #2: Square Root of a Perfect Square

The square root of a variable raised to an even power equals the variable raised to ½ that power.

• Rule # 3 – Quotient Rule for Square Roots

0a b a b a b provided a o and b

2 0n na a a

0 0a a provided a and bbb

HOMEWORK 9.4

Page 559 - 560:

# 13, 15, 19, 23, 31, 37, 41, 43, 61, 71