9.4 Multiplying and Dividing Radicals

9.4

Multiplying and Dividing Radicals

9.4

OBJECTIVES

1. Multiply and divide expressions involving numeric

radicals

2. Multiply and divide expressions involving

algebraic radicals

In Section 9.2 we stated the first property for radicals:

1ab  1a  1b

when a and b are any positive real numbers

That property has been used to simplify radical expressions up to this point. Suppose now

that we want to find a product, such as 13  15.

We can use our first radical rule in the opposite manner.

NOTE The product of square

1a  1b  1ab

roots is equal to the square root

of the product of the radicands.

so

13  15  13  5  115

We may have to simplify after multiplying, as Example 1 illustrates.

Example 1

Simplifying Radical Expressions

Multiply then simplify each expression.

(a) 15  110  15  10  150

 125  2  512

(b) 112  16  112  6  172

 136  2  136  12  612

An alternative approach would be to simplify 112 first.

112  16  213 16  2118

 219  2  2 19 12

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 2  312  6 12

(c) 110x  12x  220x2  24x2  5

 24x2  15  2x15

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CHAPTER 9

EXPONENTS AND RADICALS

CHECK YOURSELF 1

Simplify.

(a) 13  16

(c) 18a  13a

(b) 13  118

If coefficients are involved in a product, we can use the commutative and associative

properties to change the order and grouping of the factors. This is illustrated in Example 2.

Example 2

Multiplying Radical Expressions

Multiply.

(215)(3 16)  (2  3)(15  16)

necessary to show the

intermediate steps.

 6 15  6

 6130

CHECK YOURSELF 2

Multiply (317)(513).

The distributive property can also be applied in multiplying radical expressions.

Consider the following.

Example 3

Multiplying Radical Expressions

Multiply.

(a) 13(12  13)

 13  12  13  13

The distributive property

 16  3

Multiply the radicals.

(b) 15(216  313)

 15  216  15  313

The distributive property

 2  15  16  3  15  13

The commutative property

 2130  3 115

CHECK YOURSELF 3

Multiply.

(a) 15(16  15)

(b) 13(215  312)

The FOIL pattern we used for multiplying binomials in Section 3.4 can also be applied

in multiplying radical expressions. This is shown in Example 4.

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NOTE In practice, it is not

MULTIPLYING AND DIVIDING RADICALS

SECTION 9.4

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Example 4

Multiplying Radical Expressions

Multiply.

(a) (13  2)(13  5)

 13  13  513  213  2  5

 3  513  213  10

Combine like terms.

 13  713

C A U TI O N

NOTE You can use the pattern

(a  b)(a  b)  a2  b2, where

a  17 and b  2, for the same

result. 17  2 and 17  2 are

called conjugates of each other.

Note that their product is the

rational number 3. The product

of conjugates will always be

rational.

Be Careful! This result cannot be further simplified: 13 and 7 13 are not like terms.

(b) (17  2)(17  2)  17  17  2 17  217  4

743

(c) (13  5)2  (13  5)(13  5)

 13  13  513  513  5  5

 3  513  513  25

 28  1013

CHECK YOURSELF 4

Multiply.

(a) (15  3)(15  2)

(b) (13  4)(13  4)

(c) (12  3)2

We can also use our second property for radicals in the opposite manner.

NOTE The quotient of square

roots is equal to the square root

of the quotient of the

radicands.

1a

a



1b

Ab

One use of this property to divide radical expressions is illustrated in Example 5.

Example 5

Simplifying Radical Expressions

Simplify.

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NOTE The clue to recognizing

(a)

148

48



 116  4

13

A3

(b)

1200

200



 1100  10

12

A 2

(c)

2125x2

125x2

 225x2  5x



15

A 5

when to use this approach is in

noting that 48 is divisible by 3.

There is one final quotient form that you may encounter in simplifying expressions, and

it will be extremely important in our work with quadratic equations in the next chapter. This

form is shown in Example 6.

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CHAPTER 9

EXPONENTS AND RADICALS

CHECK YOURSELF 5

Simplify.

(a)

175

13

(b)

281s2

19

Example 6

Simplifying Radical Expressions

Simplify the expression

3  172

3

Be Careful! Students are

sometimes tempted to write

3  612

 1  612

3



First, we must simplify the radical in the numerator.

3  172

3  136  2



3

3

This is not correct. We must

divide both terms of the

numerator by the common

factor.



3  136  12

3  612



3

3



3(1  212)

 1  212

3

Use Property 1 to simplify 172.

Factor the numerator��then divide

by the common factor 3.

CHECK YOURSELF 6

Simplify

15  175

.

5

CHECK YOURSELF ANSWERS

1. (a) 312; (b) 316; (c) 2a16

2. 15121

3. (a) 130  5;

(b) 2115  3 16

4. (a) 1  15; (b) 13; (c) 11  612

5. (a) 5; (b) 3s

6. 3  13

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C A U TI ON

Name

9.4 Exercises

Section

Date

Perform the indicated multiplication. Then simplify each radical expression.

1. 17  15

2. 13  17

ANSWERS

1.

2.

3. 15  111

4. 113  15

3.

4.

5. 13  110m

6. 17a  113

5.

6.

7. 12x  115

8. 117  12b

7.

8.

9.

9. 13  1 7  12

10. 15  17  13

10.

11.

11. 13  112

12. 17  17

12.

13.

13. 110  110

14. 15  115

14.

15.

16.

15. 118  16

16. 18  110

17.

18.

17. 12x  16x

18. 13a  115a

19.

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20.

21.

19. 213  17

20. 312  15

22.

21. (313)(5 17)

22. (2 15)(3 111)

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