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1.5 Properties of Real Numbers

Commutative, associative, identity, inverse, and zero properties, plus the Distributive Property as the workhorse for simplifying expressions.

Intermediate Algebra 2e, 1.5 Properties of Real Numbers

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124 exercises

1.5 Exercise 1

Simplify: 18p+6q+15p+5q.18 p + 6 q + 15 p + 5 q .

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1.5 Exercise 2

Simplify: 23r+14s+9r+15s.23 r + 14 s + 9 r + 15 s .

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1.5 Exercise 3

Simplify:37m+21n+4m−15n.37 m + 21 n + 4 m - 15 n .

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1.5 Exercise 4

Simplify: (513+34)+14.\left(\frac{5}{13} + \frac{3}{4}\right) + \frac{1}{4} .

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1.5 Exercise 5

Simplify: (715+58)+38.\left(\frac{7}{15} + \frac{5}{8}\right) + \frac{3}{8} .

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1.5 Exercise 6

Simplify: (29+712)+512.\left(\frac{2}{9} + \frac{7}{12}\right) + \frac{5}{12} .

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1.5 Exercise 7

Simplify: −84n+(−73n)+84n.−84 n + \left(−73 n\right) + 84 n .

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1.5 Exercise 8

Simplify: −27a+(−48a)+27a.−27 a + \left(−48 a\right) + 27 a .

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1.5 Exercise 9

Simplify: 39x+(−92x)+(−39x).39 x + \left(−92 x\right) + \left(−39 x\right) .

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1.5 Exercise 10

Simplify: 715·823·157.\frac{7}{15} \cdot \frac{8}{23} \cdot \frac{15}{7} .

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1.5 Exercise 11

Simplify: 916·549·169.\frac{9}{16} \cdot \frac{5}{49} \cdot \frac{16}{9} .

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1.5 Exercise 12

Simplify: 617·1125·176.\frac{6}{17} \cdot \frac{11}{25} \cdot \frac{17}{6} .

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1.5 Exercise 13

Simplify: ⓐ 0n+5,\frac{0}{n + 5} , where n≠−5n \neq − 5 ⓑ 10−3p0,\frac{10 - 3 p}{0} , where 10−3p≠0.10 - 3 p \neq 0 .

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1.5 Exercise 14

Simplify: ⓐ 0m+7,\frac{0}{m + 7} , where m≠−7m \neq − 7 ⓑ 18−6c0,\frac{18 - 6 c}{0} , where 18−6c≠0.18 - 6 c \neq 0 .

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1.5 Exercise 15

Simplify: ⓐ 0d−4,\frac{0}{d - 4} , where d≠4d \neq 4 ⓑ 15−4q0,\frac{15 - 4 q}{0} , where 15−4q≠0.15 - 4 q \neq 0 .

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1.5 Exercise 16

Simplify: 3(x+4).3 \left(x + 4\right) .

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1.5 Exercise 17

Simplify: 4(x+2).4 \left(x + 2\right) .

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1.5 Exercise 18

Simplify: 6(x+7).6 \left(x + 7\right) .

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1.5 Exercise 20

Simplify: 6(56y+12).6 \left(\frac{5}{6} y + \frac{1}{2}\right) .

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1.5 Exercise 21

Simplify: 12(13n+34).12 \left(\frac{1}{3} n + \frac{3}{4}\right) .

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1.5 Exercise 23

Simplify: 100(0.7+0.15p).100 \left(0.7 + 0.15 p\right) .

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1.5 Exercise 24

Simplify: 100(0.04+0.35d).100 \left(0.04 + 0.35 d\right) .

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1.5 Exercise 25

Simplify: −11(4−3a).−11 \left(4 - 3 a\right) .

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1.5 Exercise 26

Simplify: −5(2−3a).−5 \left(2 - 3 a\right) .

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1.5 Exercise 27

Simplify: −7(8−15y).−7 \left(8 - 15 y\right) .

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1.5 Exercise 28

Simplify: −(y+5).− \left(y + 5\right) .

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1.5 Exercise 29

Simplify: −(z−11).− \left(z - 11\right) .

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1.5 Exercise 30

Simplify: −(x−4).− \left(x - 4\right) .

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1.5 Exercise 31

Simplify: 8−2(x+3)8 - 2 \left(x + 3\right)

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1.5 Exercise 32

Simplify: 9−3(x+2).9 - 3 \left(x + 2\right) .

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1.5 Exercise 33

Simplify: 7x−5(x+4).7 x - 5 \left(x + 4\right) .

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1.5 Exercise 34

Simplify: 4(x−8)−(x+3).4 \left(x - 8\right) - \left(x + 3\right) .

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1.5 Exercise 35

Simplify: 6(x−9)−(x+12).6 \left(x - 9\right) - \left(x + 12\right) .

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1.5 Exercise 36

Simplify: 8(x−1)−(x+5).8 \left(x - 1\right) - \left(x + 5\right) .

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1.5 Exercise 37

Simplify.

43m+(−12n)+(−16m)+(−9n)43 m + \left(−12 n\right) + \left(−16 m\right) + \left(−9 n\right)

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1.5 Exercise 39

Simplify.

38g+112h+78g+512h\frac{3}{8} g + \frac{1}{12} h + \frac{7}{8} g + \frac{5}{12} h

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1.5 Exercise 41

Simplify.

6.8p+9.14q+(−4.37p)+(−0.88q)6.8 p + 9.14 q + \left(−4.37 p\right) + \left(−0.88 q\right)

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1.5 Exercise 43

Simplify.

−24·7·38−24 \cdot 7 \cdot \frac{3}{8}

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1.5 Exercise 45

Simplify.

(56+815)+715\left(\frac{5}{6} + \frac{8}{15}\right) + \frac{7}{15}

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1.5 Exercise 47

Simplify.

17(0.25)(4)17 \left(0.25\right) \left(4\right)

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1.5 Exercise 49

Simplify.

[2.48(12)](0.5)\left[2.48 \left(12\right)\right] \left(0.5\right)

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1.5 Exercise 51

Simplify.

12(56p)12 \left(\frac{5}{6} p\right)

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1.5 Exercise 53

Simplify.

19a+44−19a19 a + 44 - 19 a

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1.5 Exercise 55

Simplify.

12+78+(−12)\frac{1}{2} + \frac{7}{8} + \left(- \frac{1}{2}\right)

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1.5 Exercise 57

Simplify.

10(0.1d)10 \left(0.1 d\right)

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1.5 Exercise 59

Simplify.

320·4911·203\frac{3}{20} \cdot \frac{49}{11} \cdot \frac{20}{3}

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1.5 Exercise 61

Simplify.

0u−4.99,\frac{0}{u - 4.99} , where u≠4.99u \neq 4.99

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1.5 Exercise 63

Simplify.

32−5a0,\frac{32 - 5 a}{0} , where 32−5a≠032 - 5 a \neq 0

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1.5 Exercise 65

Simplify.

(34+910m)÷0,\left(\frac{3}{4} + \frac{9}{10} m\right) \div 0 , where 34+910m≠0\frac{3}{4} + \frac{9}{10} m \neq 0

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1.5 Exercise 67

Simplify using the Distributive Property.

8(4y+9)8 \left(4 y + 9\right)

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1.5 Exercise 69

Simplify using the Distributive Property.

6(c−13)6 \left(c - 13\right)

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1.5 Exercise 71

Simplify using the Distributive Property.

14(3q+12)\frac{1}{4} \left(3 q + 12\right)

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1.5 Exercise 73

Simplify using the Distributive Property.

9(59y−13)9 \left(\frac{5}{9} y - \frac{1}{3}\right)

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1.5 Exercise 75

Simplify using the Distributive Property.

12(14+23r)12 \left(\frac{1}{4} + \frac{2}{3} r\right)

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1.5 Exercise 77

Simplify using the Distributive Property.

15·35(4d+10)15 \cdot \frac{3}{5} \left(4 d + 10\right)

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1.5 Exercise 79

Simplify using the Distributive Property.

r(s−18)r \left(s - 18\right)

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1.5 Exercise 81

Simplify using the Distributive Property.

(y+4)p\left(y + 4\right) p

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1.5 Exercise 83

Simplify using the Distributive Property.

−7(4p+1)−7 \left(4 p + 1\right)

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1.5 Exercise 85

Simplify using the Distributive Property.

−3(x−6)−3 \left(x - 6\right)

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1.5 Exercise 87

Simplify using the Distributive Property.

−(3x−7)− \left(3 x - 7\right)

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1.5 Exercise 89

Simplify using the Distributive Property.

16−3(y+8)16 - 3 \left(y + 8\right)

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1.5 Exercise 91

Simplify using the Distributive Property.

4−11(3c−2)4 - 11 \left(3 c - 2\right)

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1.5 Exercise 93

Simplify using the Distributive Property.

22−(a+3)22 - \left(a + 3\right)

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1.5 Exercise 95

Simplify using the Distributive Property.

(5m−3)−(m+7)\left(5 m - 3\right) - \left(m + 7\right)

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1.5 Exercise 97

Simplify using the Distributive Property.

9(8x−3)−(−2)9 \left(8 x - 3\right) - \left(−2\right)

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1.5 Exercise 99

Simplify using the Distributive Property.

5(2n+9)+12(n−3)5 \left(2 n + 9\right) + 12 \left(n - 3\right)

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1.5 Exercise 101

Simplify using the Distributive Property.

14(c−1)−8(c−6)14 \left(c - 1\right) - 8 \left(c - 6\right)

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1.5 Exercise 103

Simplify using the Distributive Property.

6(7y+8)−(30y−15)6 \left(7 y + 8\right) - \left(30 y - 15\right)

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1.5 Exercise 105

In your own words, state the Associative Property of addition.

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1.5 Exercise 107

Simplify 8(x−14)8 \left(x - \frac{1}{4}\right) using the Distributive Property and explain each step.

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1.5 Exercise 109

Identify Multiples and Factors

Use the divisibility tests to determine whether 180 is divisible by 2, by 3, by 5, by 6, and by 10.

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1.5 Exercise 111

Identify Multiples and Factors

Find the least common multiple of 24 and 40.

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1.5 Exercise 113

Simplify each expression.

7+3[6−4(5−4)]−327 + 3 \left[6 - 4 \left(5 - 4\right)\right] - 3^{2}

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1.5 Exercise 115

Evaluate the following expressions.

2x2−4xy−3y22 x^{2} - 4 x y - 3 y^{2} when x=3,x = 3 , y=1y = 1

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1.5 Exercise 117

Simplify the following expressions by combining like terms.

14x2−9x+11−8x2+8x−614 x^{2} - 9 x + 11 - 8 x^{2} + 8 x - 6

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1.5 Exercise 119

Translate the phrases into algebraic expressions.


ⓐ eleven times the difference of yy and two
ⓑ the difference of eleven times yy and two

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1.5 Exercise 121

In the following exercise, fill in <,>,< , > , or == for each of the following pairs of numbers.


ⓐ −|7|___−|−7|− \left|7\right| ___ - \left|−7\right|
ⓑ −8___−|−8|−8 ___ - \left|−8\right|
ⓒ |−13|___−13\left|−13\right| ___ - 13
ⓓ |−12|___−(−12)\left|−12\right| ___ - \left(−12\right)

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1.5 Exercise 123

Simplify.

12−3|1−4(4−2)|12 - 3 \left|1 - 4 \left(4 - 2\right)\right|

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1.5 Exercise 125

Simplify each expression.


ⓐ 15−715 - 7
ⓑ −15−(−7)−15 - \left(−7\right)
ⓒ −15−7−15 - 7
ⓓ 15−(−7)15 - \left(−7\right)

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1.5 Exercise 127

Simplify each expression.

ⓐ 23−(−17)23 - \left(−17\right) ⓑ 23+1723 + 17

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1.5 Exercise 129

In the following exercise, multiply or divide.

ⓐ −27÷9−27 \div 9 ⓑ 120÷(−8)120 \div \left(−8\right) ⓒ 4(−14)4 \left(−14\right) ⓓ −1(−17)−1 \left(−17\right)

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1.5 Exercise 131

Simplify each expression.

(7−11)(6−13)\left(7 - 11\right) \left(6 - 13\right)

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1.5 Exercise 133

Simplify each expression.

6−3|4(1−2)−(7−5)|6 - 3 \left|4 \left(1 - 2\right) - \left(7 - 5\right)\right|

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1.5 Exercise 135

Evaluate each expression.

(y+z)2\left(y + z\right)^{2} when
y=−4,z=7y = −4 , z = 7

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1.5 Exercise 137

Translate to an algebraic expression and simplify if possible.

the sum of −4−4 and −9,−9 , increased by 23

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1.5 Exercise 139

In the following exercise, solve.

Temperature On July 10, the high temperature in Phoenix, Arizona, was 109°, and the high temperature in Juneau, Alaska, was 63°. What was the difference between the temperature in Phoenix and the temperature in Juneau?

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1.5 Exercise 141

Simplify.

−270x3198y2- \frac{270 x^{3}}{198 y^{2}}

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1.5 Exercise 143

Perform the indicated operation.

6x25÷9y20\frac{6 x}{25} \div \frac{9 y}{20}

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1.5 Exercise 145

Perform the indicated operation.

518+712\frac{5}{18} + \frac{7}{12}

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1.5 Exercise 147

Perform the indicated operation.

ⓐ 58+34\frac{5}{8} + \frac{3}{4} ⓑ 58÷34\frac{5}{8} \div \frac{3}{4}

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1.5 Exercise 149

Simplify.

4·3−2·5−6·3+2·3\frac{4 \cdot 3 - 2 \cdot 5}{−6 \cdot 3 + 2 \cdot 3}

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1.5 Exercise 151

Simplify.

43−42(45)2\frac{4^{3} - 4^{2}}{\left(\frac{4}{5}\right)^{2}}

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1.5 Exercise 153

Evaluate.

a+ba−b\frac{a + b}{a - b} when
a=−4,a = −4 , b=6b = 6

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1.5 Exercise 155

Perform the indicated operation.

−23.67+29.84−23.67 + 29.84

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1.5 Exercise 157

Perform the indicated operation.

79.38−(−17.598)79.38 - \left(−17.598\right)

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1.5 Exercise 159

Perform the indicated operation.

(−8.43)(−57.91)\left(−8.43\right) \left(−57.91\right)

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1.5 Exercise 161

Perform the indicated operation.

(0.563)(100)\left(0.563\right) \left(100\right)

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1.5 Exercise 163

Perform the indicated operation.

1.84÷(−0.8)1.84 \div \left(−0.8\right)

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1.5 Exercise 165

Convert each decimal to a fraction.

−9.6−9.6

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1.5 Exercise 167

Convert each fraction to a decimal.

1411\frac{14}{11}

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1.5 Exercise 169

Convert each decimal to a percent.

0.04750.0475

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1.5 Exercise 171

Simplify.

−121\sqrt{−121}

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1.5 Exercise 175

Simplify.

58x+512y+18x+712y\frac{5}{8} x + \frac{5}{12} y + \frac{1}{8} x + \frac{7}{12} y

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1.5 Exercise 177

Simplify.

(1115+38)+58\left(\frac{11}{15} + \frac{3}{8}\right) + \frac{5}{8}

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1.5 Exercise 179

Simplify.

1315·917·1513\frac{13}{15} \cdot \frac{9}{17} \cdot \frac{15}{13}

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1.5 Exercise 181

Simplify.

5x−70,5x−7≠0\frac{5 x - 7}{0} , 5 x - 7 \neq 0

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1.5 Exercise 183

Simplify using the Distributive Property.

12(23b+56)12 \left(\frac{2}{3} b + \frac{5}{6}\right)

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1.5 Exercise 185

Simplify using the Distributive Property.

(x−5)p\left(x - 5\right) p

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1.5 Exercise 187

Simplify using the Distributive Property.

12−6(x+3)12 - 6 \left(x + 3\right)

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1.5 Exercise 189

Simplify using the Distributive Property.

5(2y+3)−(4y−1)5 \left(2 y + 3\right) - \left(4 y - 1\right)

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1.5 Exercise 191

Combine like terms: 5n+8+2n−15 n + 8 + 2 n - 1

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1.5 Exercise 193

Translate to an algebraic expression and simplify:

ⓐ eleven less than negative eight

ⓑ the difference of −8−8 and −3−3, increased by 5

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1.5 Exercise 195

Round 28.145828.1458 to the nearest

ⓐ hundredth ⓑ thousandth

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1.5 Exercise 199

Simplify each expression.

−(4−9)−(9−5)− \left(4 - 9\right) - \left(9 - 5\right)

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1.5 Exercise 201

Simplify each expression.

16−2|3(1−4)−(8−5)|16 - 2 \left|3 \left(1 - 4\right) - \left(8 - 5\right)\right|

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1.5 Exercise 203

Simplify each expression.

180204\frac{180}{204}

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1.5 Exercise 205

Simplify each expression.

45÷(−1225)\frac{4}{5} \div \left(- \frac{12}{25}\right)

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1.5 Exercise 207

Simplify each expression.

4(−3+2(3−6))3(11−3(2+3))\frac{4 \left(−3 + 2 \left(3 - 6\right)\right)}{3 \left(11 - 3 \left(2 + 3\right)\right)}

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1.5 Exercise 209

Simplify each expression.

−591021\frac{- \frac{5}{9}}{\frac{10}{21}}

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1.5 Exercise 211

Simplify each expression.

34.6−10034.6 - 100

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1.5 Exercise 213

Simplify each expression.

−8÷0.05−8 \div 0.05

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1.5 Exercise 215

Simplify each expression.

(813+57)+27\left(\frac{8}{13} + \frac{5}{7}\right) + \frac{2}{7}

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1.5 Exercise 217

Simplify each expression.

ⓐ 09\frac{0}{9} ⓑ 110\frac{11}{0}

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1.5 Exercise 219

Simplify each expression.

6(3y−1)−(5y−3)6 \left(3 y - 1\right) - \left(5 y - 3\right)

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Adapted from Intermediate Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Intermediate Algebra 2e. Imported at revision 77a933c.