multiplying and dividing radicals the product and quotient properties of square roots can be used to...

Multiplying and Dividing Radicals

The product and quotient properties of square roots can be used to multiply and divide radicals, because:

baba andb

a

b

a .

Example 1:

82 82 16 4

Example 2:

6

5

15

2

615

52

9

1)32()35(

52

3

1

Example 3:

6332 63 )32( 18 6 29 6 2 18

Product Rule

• Simplify radicals

• Multiply Coefficients• Multiply radicands

– “Roots” must be the same

• Simplify, if needed

4827

3933

12 9

31263

316

34

Examples: Product Rule

98824924

2722

417217

34

1756372579

7573

4915715

105

Quotient Rule

• Fractions made up of radicals can be simplified just like fractions

1456

1456

4 2

xyyx

315 3

xyyx

315 3

25x

52 x

5x

73

97

375

yxyx

73

97

375

yxyx

2425 yx

yx25

Multiply the radicals.

1. Simplify. 6 10

60 2 15

Simplify: 60

60 4 g 15

6. Simplify.

Multiply the coefficients and the radicals.

2 14 3 21

42 6

6 294

6 49 g 6

6 g 7 6

7. Simplify.

Divide the radicals.

108

3

108

3

36

6

8. Simplify.

8 2

2 8

4

22

8 2

2 8

8

2

2

44

1

44

1

4

Rationalizing Radicals

To simplify a fraction with a radical in the denominator, multiply the numerator and denominator by the radical.

Example 1:

2

1

2

2

2

2 Estimation is easier with rational denominators.

This process is called rationalizing the denominator.Example 2:

3

2

3

3

3

2

3

6

Since the square root of a quotient is a quotient of square roots, the square root of a fraction must be rationalized to be in simplest form.

Answer:

9. Simplify.

5

7

35

7

Radicals representing square roots of different numbers can not be gathered like this.

Adding and Subtracting Radicals

Radicals that represent the square root of the same number can be treated as a common factor.

Examples:

3 2 3 4

2 2 2 5

But simplifying sometimes results in multiples of the same radical, which can be.

Examples:

12234 34234 383)22(34

2055 5455 5255 53

Like terms can be gathered. Unlike terms can not.

3 )24(

2 3

3 6

2 )25(

Combining Like Terms

• Radicals & Like Terms– Same variables– Variables have the same exponents– IDENTICAL RADICALS

• Examplesxx 32&34 xy

xxyx 2

32

&252

2

• Simplify radicals if possible

• Combine coefficients

3534

Radicals ARE simplified

3

1. Simplify.

Just like when adding variables, you can only combine LIKE radicals.

5 √5

3 5 4 5 2 5

Answer: 4 √7 +3 √3

2. Simplify. 6 7 3 2 7 4 3

Simplify each radical.4√9•3 - 2√16 • 3 + 2√4•54 • 3√3 - 2 • 4√3+2 • 2√5

12√3 - 8√3 + 4√5Combine like radicals

4√3 + 4√5

3. Simplify. 4 27 2 48 2 20

More Radical Fun

103

210

2

1

64

63

1067

6438

64322

6462

62

SIMPLIFY

MULTIPLY

Must have Common Denominators

1010

Distributive Property with Radicals

)32(2 22

64

62

)325274(32

)3(2 8 81 10 96

8 9 10 616

72 641064072

Multiplying Binomials With Radicals

Multiplying binomials that contain radicals sometimes results in products of radicals that can be simplified.

Examples:22 )5()3( 9 - 5 4)53( )53( 1.

2)53( 2. 2)5(569 5614

3. 2)3423( 22 3 4 3 4 2 3 2 2 3

)3(16624)2(9

62466

4862418

Conjugates

Binomials of the form and dcba dcba that are identical except for the sign separating the terms are calledconjugates.

Multiplying conjugates like these together results in a rational number:

dcba dcba

Conjugates are therefore used to rationalize certain fractions.

Example:

2222 d c b a a2b - c2d

223

4

223

223

2 22 9

223 4

)2(49

2812

89

2812

2812

Practice

Multiply: 632

Divide:6

32

Add: 1232

Subtract: 1823

Multiply: 323 323

Rationalize: )323(

3

26

15

2

34

0

5

323