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Simplify expressions using the properties of identities, inverses, and zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

Simplify: 3 x + 15 3 x .

Solution

3 x + 15 3 x
Notice the additive inverses, 3 x and −3 x . 0 + 15
Add. 15
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Simplify: −12 z + 9 + 12 z .

9

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Simplify: −25 u 18 + 25 u .

−18

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Simplify: 4 ( 0.25 q ) .

Solution

4 ( 0.25 q )
Regroup, using the associative property. [ 4 ( 0.25 ) ] q
Multiply. 1.00 q
Simplify; 1 is the multiplicative identity. q
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Simplify: 2 ( 0.5 p ) .

p

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Simplify: 25 ( 0.04 r ) .

r

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Simplify: 0 n + 5 , where n −5 .

Solution

0 n + 5
Zero divided by any real number except itself is zero. 0
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Simplify: 0 m + 7 , where m −7 .

0

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Simplify: 0 d 4 , where d 4 .

0

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Simplify: 10 3 p 0 .

Solution

10 3 p 0
Division by zero is undefined. undefined
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Simplify: 18 6 c 0 .

undefined

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Simplify: 15 4 q 0 .

undefined

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Simplify: 3 4 · 4 3 ( 6 x + 12 ) .

Solution

We cannot combine the terms in parentheses, so we multiply the two fractions first.

3 4 · 4 3 ( 6 x + 12 )
Multiply; the product of reciprocals is 1. 1 ( 6 x + 12 )
Simplify by recognizing the multiplicative identity. 6 x + 12
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Simplify: 2 5 · 5 2 ( 20 y + 50 ) .

20 y + 50

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Simplify: 3 8 · 8 3 ( 12 z + 16 ) .

12 z + 16

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All the properties of real numbers we have used in this chapter are summarized in [link] .

Properties of real numbers
Property Of Addition Of Multiplication
Commutative Property
If a and b are real numbers then… a + b = b + a a · b = b · a
Associative Property
If a , b , and c are real numbers then… ( a + b ) + c = ( b + c ) ( a · b ) · c = a · ( b · c )
Identity Property 0 is the additive identity 1 is the multiplicative identity
For any real number a , a + 0 = a 0 + a = a a · 1 = a 1 · a = a
Inverse Property a is the additive inverse of a a , a 0
1 a is the multiplicative inverse of a
For any real number a , a + ( a ) = 0 a · 1 a = 1
Distributive Property
If a , b , c are real numbers, then a ( b + c ) = a b + a c
Properties of Zero
For any real number a ,
a 0 = 0 0 a = 0
For any real number a , a 0 0 a = 0
a 0 is undefined

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Key concepts

  • Identity Properties
    • Identity Property of Addition: For any real number a : a + 0 = a 0 + a = a 0 is the additive identity
    • Identity Property of Multiplication: For any real number a : a 1 = a 1 a = a 1 is the multiplicative identity
  • Inverse Properties
    • Inverse Property of Addition: For any real number a : a + ( - a ) = 0 - a is the additive inverse of a
    • Inverse Property of Multiplication: For any real number a : ( a 0 ) a 1 a = 1 1 a is the multiplicative inverse of a
  • Properties of Zero
    • Multiplication by Zero: For any real number a , a 0 = 0 0 a = 0 The product of any number and 0 is 0.
    • Division of Zero: For any real number a , 0 a = 0 0 + a = 0 Zero divided by any real number, except itself, is zero.
    • Division by Zero: For any real number a, 0 a is undefined and a ÷ 0 is undefined. Division by zero is undefined.

Practice makes perfect

Recognize the Identity Properties of Addition and Multiplication

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

3 5 ( 1 ) = 3 5

identity property of multiplication

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0 + 64 = 64

identity property of addition

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Use the Inverse Properties of Addition and Multiplication

In the following exercises, find the multiplicative inverse.

Use the Properties of Zero

In the following exercises, simplify using the properties of zero.

Simplify Expressions using the Properties of Identities, Inverses, and Zero

In the following exercises, simplify using the properties of identities, inverses, and zero.

0 r + 20 , where r −20

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0 s + 13 , where s −13

0

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0 u 4.99 , where u 4.99

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0 v 65.1 , where v 65.1

0

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0 ÷ ( x 1 2 ) , where x 1 2

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0 ÷ ( y 1 6 ) , where y 1 6

0

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32 5 a 0 , where 32 5 a 0

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28 9 b 0 , where 28 9 b 0

undefined

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2.1 + 0.4 c 0 , where 2.1 + 0.4 c 0

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1.75 + 9 f 0 , where 1.75 + 9 f 0

undefined

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( 3 4 + 9 10 m ) ÷ 0 , where 3 4 + 9 10 m 0

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( 5 16 n 3 7 ) ÷ 0 , where 5 16 n 3 7 0

undefined

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9 10 · 10 9 ( 18 p 21 )

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5 7 · 7 5 ( 20 q 35 )

20 q − 35

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18 · 5 6 ( 15 h + 24 )

225 h + 360

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Everyday math

Insurance copayment Carrie had to have 5 fillings done. Each filling cost $80 . Her dental insurance required her to pay 20 % of the cost. Calculate Carrie’s cost

by finding her copay for each filling, then finding her total cost for 5 fillings, and

by multiplying 5 ( 0.20 ) ( 80 ) .

Which of the Properties of Real Numbers did you use for part (b)?

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Cooking time Helen bought a 24-pound turkey for her family’s Thanksgiving dinner and wants to know what time to put the turkey in the oven. She wants to allow 20 minutes per pound cooking time.

Calculate the length of time needed to roast the turkey by multiplying 24 · 20 to find the number of minutes and then multiplying the product by 1 60 to convert minutes into hours.

Multiply 24 ( 20 · 1 60 ) .

Which of the Properties of Real Numbers allows you to multiply 24 ( 20 · 1 60 ) instead of ( 24 · 20 ) 1 60 ?

  1. 8 hours
  2. 8
  3. associative property of multiplication

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Writing exercises

In your own words, describe the difference between the additive inverse and the multiplicative inverse of a number.

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How can the use of the properties of real numbers make it easier to simplify expressions?

Answers will vary.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Practice Key Terms 4

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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