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Multiplication of Square Root Expressions

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy. Objectives of this module: be able to use the product property of square roots to multiply square roots.

Overview

  • The Product Property of Square Roots
  • Multiplication Rule for Square Root Expressions

The Product Property of Square Roots

In our work with simplifying square root expressions, we noted that

xy = x y xy = x y

Since this is an equation, we may write it as

x y = xy x y = xy

To multiply two square root expressions, we use the product property of square roots.

The Product Property x y = xy x y = xy

x y = xy x y = xy

The product of the square roots is the square root of the product.

In practice, it is usually easier to simplify the square root expressions before actually performing the multiplication. To see this, consider the following product:

8 48 8 48
We can multiply these square roots in either of two ways:

Example 1

Simplify then multiply.

4·2 16·3 =( 2 2 )( 4 3 )=2·4 2·3 =8 6 4·2 16·3 =( 2 2 )( 4 3 )=2·4 2·3 =8 6

Example 2

Multiply then simplify.

8 48 = 8·48 = 384 = 64·6 =8 6 8 48 = 8·48 = 384 = 64·6 =8 6

Notice that in the second method, the expanded term (the third expression, 384 384 ) may be difficult to factor into a perfect square and some other number.

Multiplication Rule for Square Root Expressions

The preceding example suggests that the following rule for multiplying two square root expressions.

Rule for Multiplying Square Root Expressions

  1. Simplify each square root expression, if necessary.
  2. Perform the multiplecation.
  3. Simplify, if necessary.

Sample Set A

Find each of the following products.

Example 3

3 6 = 3·6 = 18 = 9·2 =3 2 3 6 = 3·6 = 18 = 9·2 =3 2

Example 4

8 2 =2 2 2 =2 2·2 =2 4 =2·2=4 8 2 =2 2 2 =2 2·2 =2 4 =2·2=4

This product might be easier if we were to multiply first and then simplify.

8 2 = 8·2 = 16 =4 8 2 = 8·2 = 16 =4

Example 5

20 7 = 4 5 7 =2 5·7 =2 35 20 7 = 4 5 7 =2 5·7 =2 35

Example 6

5 a 3 27 a 5 =(a 5a )(3 a 2 3a ) = 3 a 3 15 a 2 = 3 a 3 ·a 15 = 3 a 4 15 5 a 3 27 a 5 =(a 5a )(3 a 2 3a ) = 3 a 3 15 a 2 = 3 a 3 ·a 15 = 3 a 4 15

Example 7

(x+2) 7 x−1 = (x+2) 6 (x+2) x−1 = (x+2) 3 (x+2) x−1 = (x+2) 3 (x+2)(x−1) or = (x+2) 3 x 2 +x−2 (x+2) 7 x−1 = (x+2) 6 (x+2) x−1 = (x+2) 3 (x+2) x−1 = (x+2) 3 (x+2)(x−1) or = (x+2) 3 x 2 +x−2

Example 8

Finding the product of the square root of three and the binomial seven plus the square root of six, using the rule for multiplying square root expressions. See the longdesc for a full description.

Example 9

Finding the product of the square root of six and the binomial the square root of two minus the square root of ten, using the rule for multiplying square root expressions. See the longdesc for a full description.

Example 10

Finding the product of two expressions involving square roots. The first expression is the square root of forty-five a to the sixth power b cubed. The second expression is a binomial with the square root of 'five times the cube of the binomial b minus three. See the longdesc for a full description.

Practice Set A

Find each of the following products.

Exercise 1

5 6 5 6

Solution

30 30

Exercise 2

32 2 32 2

Solution

8

Exercise 3

x+4 x+3 x+4 x+3

Solution

( x+4 )( x+3 ) ( x+4 )( x+3 )

Exercise 4

8 m 5 n 20 m 2 n 8 m 5 n 20 m 2 n

Solution

4 m 3 n 10m 4 m 3 n 10m

Exercise 5

9 ( k−6 ) 3 k 2 −12k+36 9 ( k−6 ) 3 k 2 −12k+36

Solution

3 ( k−6 ) 2 k−6 3 ( k−6 ) 2 k−6

Exercise 6

3 ( 2 + 5 ) 3 ( 2 + 5 )

Solution

6 + 15 6 + 15

Exercise 7

2a ( 5a − 8 a 3 ) 2a ( 5a − 8 a 3 )

Solution

a 10 −4 a 2 a 10 −4 a 2

Exercise 8

32 m 5 n 8 ( 2m n 2 − 10 n 7 ) 32 m 5 n 8 ( 2m n 2 − 10 n 7 )

Solution

8 m 3 n 2 n −8 m 2 n 5 5m 8 m 3 n 2 n −8 m 2 n 5 5m

Exercises

Exercise 9

2 10 2 10

Solution

2 5 2 5

Exercise 10

3 15 3 15

Exercise 11

7 8 7 8

Solution

2 14 2 14

Exercise 12

20 3 20 3

Exercise 13

32 27 32 27

Solution

12 6 12 6

Exercise 14

45 50 45 50

Exercise 15

5 5 5 5

Solution

5

Exercise 16

7 7 7 7

Exercise 17

8 8 8 8

Solution

8

Exercise 18

15 15 15 15

Exercise 19

48 27 48 27

Solution

36

Exercise 20

80 20 80 20

Exercise 21

5 m 5 m

Solution

5m 5m

Exercise 22

7 a 7 a

Exercise 23

6 m 6 m

Solution

6m 6m

Exercise 24

10 h 10 h

Exercise 25

20 a 20 a

Solution

2 5a 2 5a

Exercise 26

48 x 48 x

Exercise 27

75 y 75 y

Solution

5 3y 5 3y

Exercise 28

200 m 200 m

Exercise 29

a a a a

Solution

a a

Exercise 30

x x x x

Exercise 31

y y y y

Solution

y y

Exercise 32

h h h h

Exercise 33

3 3 3 3

Solution

3

Exercise 34

6 6 6 6

Exercise 35

k k k k

Solution

k k

Exercise 36

m m m m

Exercise 37

m 2 m m 2 m

Solution

m m m m

Exercise 38

a 2 a a 2 a

Exercise 39

x 3 x x 3 x

Solution

x 2 x 2

Exercise 40

y 3 y y 3 y

Exercise 41

y y 4 y y 4

Solution

y 2 y y 2 y

Exercise 42

k k 6 k k 6

Exercise 43

a 3 a 5 a 3 a 5

Solution

a 4 a 4

Exercise 44

x 3 x 7 x 3 x 7

Exercise 45

x 9 x 3 x 9 x 3

Solution

x 6 x 6

Exercise 46

y 7 y 9 y 7 y 9

Exercise 47

y 3 y 4 y 3 y 4

Solution

y 3 y y 3 y

Exercise 48

x 8 x 5 x 8 x 5

Exercise 49

x+2 x−3 x+2 x−3

Solution

( x+2 )( x−3 ) ( x+2 )( x−3 )

Exercise 50

a−6 a+1 a−6 a+1

Exercise 51

y+3 y−2 y+3 y−2

Solution

( y+3 )( y−2 ) ( y+3 )( y−2 )

Exercise 52

h+1 h−1 h+1 h−1

Exercise 53

x+9 ( x+9 ) 2 x+9 ( x+9 ) 2

Solution

( x+9 ) x+9 ( x+9 ) x+9

Exercise 54

y−3 ( y−3 ) 5 y−3 ( y−3 ) 5

Exercise 55

3 a 2 15 a 3 3 a 2 15 a 3

Solution

3 a 2  5a 3 a 2  5a

Exercise 56

2 m 4 n 3 14 m 5 n 2 m 4 n 3 14 m 5 n

Exercise 57

12 ( p−q ) 3 3 ( p−q ) 5 12 ( p−q ) 3 3 ( p−q ) 5

Solution

6 ( p−q ) 4 6 ( p−q ) 4

Exercise 58

15 a 2 ( b+4 ) 4 21 a 3 ( b+4 ) 5 15 a 2 ( b+4 ) 4 21 a 3 ( b+4 ) 5

Exercise 59

125 m 5 n 4 r 8 8 m 6 r 125 m 5 n 4 r 8 8 m 6 r

Solution

10 m 5 n 2 r 4 10mr 10 m 5 n 2 r 4 10mr

Exercise 60

7 ( 2k−1 ) 11 ( k+1 ) 3 14 ( 2k−1 ) 10 7 ( 2k−1 ) 11 ( k+1 ) 3 14 ( 2k−1 ) 10

Exercise 61

y 3 y 5 y 2 y 3 y 5 y 2

Solution

y 5 y 5

Exercise 62

x 6 x 2 x 9 x 6 x 2 x 9

Exercise 63

2 a 4 5 a 3 2 a 7 2 a 4 5 a 3 2 a 7

Solution

2 a 7 5 2 a 7 5

Exercise 64

x n x n x n x n

Exercise 65

y 2 n y 4 n y 2 n y 4 n

Solution

y 3n y 3n

Exercise 66

a 2n+5 a 3 a 2n+5 a 3

Exercise 67

2 m 3n+1 10 m n+3 2 m 3n+1 10 m n+3

Solution

2 m 2n+2 5 2 m 2n+2 5

Exercise 68

75 ( a−2 ) 7 48a−96 75 ( a−2 ) 7 48a−96

Exercise 69

2 ( 8 + 6 ) 2 ( 8 + 6 )

Solution

2( 2+ 3 ) 2( 2+ 3 )

Exercise 70

5 ( 3 + 7 ) 5 ( 3 + 7 )

Exercise 71

3 ( x + 2 ) 3 ( x + 2 )

Solution

3x + 6 3x + 6

Exercise 72

11 ( y + 3 ) 11 ( y + 3 )

Exercise 73

8 ( a − 3a ) 8 ( a − 3a )

Solution

2 2a −2 6a 2 2a −2 6a

Exercise 74

x ( x 3 − 2 x 4 ) x ( x 3 − 2 x 4 )

Exercise 75

y ( y 5 + 3 y 3 ) y ( y 5 + 3 y 3 )

Solution

y 2 ( y+ 3 ) y 2 ( y+ 3 )

Exercise 76

8 a 5 ( 2a − 6 a 11 ) 8 a 5 ( 2a − 6 a 11 )

Exercise 77

12 m 3 ( 6 m 7 − 3m ) 12 m 3 ( 6 m 7 − 3m )

Solution

6 m 2 ( m 3 2 −1 ) 6 m 2 ( m 3 2 −1 )

Exercise 78

5 x 4 y 3 ( 8xy −5 7x ) 5 x 4 y 3 ( 8xy −5 7x )

Exercises for Review

Exercise 79

((Reference)) Factor a 4 y 4 −25 w 2 . a 4 y 4 −25 w 2 .

Solution

( a 2 y 2 +5w )( a 2 y 2 −5w ) ( a 2 y 2 +5w )( a 2 y 2 −5w )

Exercise 80

((Reference)) Find the slope of the line that passes through the points ( −5,4 ) ( −5,4 ) and ( −3,4 ). ( −3,4 ).

Exercise 81

((Reference)) Perform the indicated operations:

15 x 2 −20x 6 x 2 +x−12 · 8x+12 x 2 −2x−15 ÷ 5 x 2 +15x x 2 −25 15 x 2 −20x 6 x 2 +x−12 · 8x+12 x 2 −2x−15 ÷ 5 x 2 +15x x 2 −25

Solution

4( x+5 ) ( x+3 ) 2 4( x+5 ) ( x+3 ) 2

Exercise 82

((Reference)) Simplify x 4 y 2 z 6 x 4 y 2 z 6 by removing the radical sign.

Exercise 83

((Reference)) Simplify 12 x 3 y 5 z 8 . 12 x 3 y 5 z 8 .

Solution

2x y 2 z 4 3xy 2x y 2 z 4 3xy

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