Chapter 1 Review Exercises
Chapter Review Exercises
Use the Language of Algebra
Identify Multiples and Factors
1. Use the divisibility tests to determine whether 180 is divisible by 2, by 3, by 5, by 6, and by 10.
- Answer
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Divisible by \(2,3,5,6\)
2. Find the prime factorization of 252.
3. Find the least common multiple of 24 and 40.
- Answer
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120
In the following exercises, simplify each expression.
4. \(24÷3+4(5−2)\)
5. \(7+3[6−4(5−4)]−3^2\)
- Answer
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4
Evaluate an Expression
In the following exercises, evaluate the following expressions.
6. When \(x=4\), ⓐ \(x^3\) ⓑ \(5x\) ⓒ \(2x^2−5x+3\)
7. \(2x^2−4xy−3y^2\) when \(x=3\) and \(y=1\)
- Answer
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3
In the following exercises, simplify the following expressions by combining like terms.
8. \(12y+7+2y−5\)
9. \(14x^2−9x+11−8x^2+8x−6\)
- Answer
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\(6x^2−x+5\)
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the phrases into algebraic expressions.
10. ⓐ the sum of \(4ab^2\) and \(7a3b24ab^2\) and \(7a^3b^2\)
ⓑ the product of \(6y^2\) and \(3y\)
ⓒ twelve more than \(5x\)
ⓓ \(5y\) less than \(8y^2\)
11. ⓐ eleven times the difference of \(y\) and two
ⓑ the difference of eleven times \(y\) and two
- Answer
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ⓐ \(11(y−2)\)
ⓑ \(11y−2\)
12. Dushko has nickels and pennies in his pocket. The number of pennies is four less than five the number of nickels. Let nn represent the number of nickels. Write an expression for the number of pennies.
Integers
Simplify Expressions with Absolute Value
In the following exercise, fill in \(<,>,\) or \(=\) for each of the following pairs of numbers.
13. ⓐ \(−|7| \_\_\_−|−7|\)
ⓑ \(−8 \_\_\_−|−8|\)
ⓒ \(|−13| \_\_\_−13\)
ⓓ \(|−12| \_\_\_−(−12)\)
- Answer
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ⓐ \(=\)
ⓑ \(=\)
ⓒ \(>\)
ⓓ \(=\)
In the following exercises, simplify.
14. \(9−|3(4−8)|\)
15. \(12−3|1−4(4−2)|\)
- Answer
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\(−9\)
Add and Subtract Integers
In the following exercises, simplify each expression.
16. \(−12+(−8)+7\)
ⓑ \(−15−(−7)\)
ⓒ \(−15−7\)
ⓓ \(15−(−7)\)
- Answer
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ⓐ \(8\)
ⓑ \(−8\)
ⓒ \(−22\)
ⓓ \(22\)
18. \(−11−(−12)+5\)
19. ⓐ \(23−(−17)\) ⓑ \(23+17\)
- Answer
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ⓐ 40 ⓑ 40
20. \(−(7−11)−(3−5)\)
Multiply and Divide Integers
In the following exercise, multiply or divide.
21. ⓐ \(−27÷9\) ⓑ \(120÷(−8)\) ⓒ \(4(−14)\) ⓓ \(−1(−17)\)
- Answer
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ⓐ \(−3\) ⓑ \(−15\) ⓒ \(−56\) ⓓ \(17\)
Simplify and Evaluate Expressions with Integers
In the following exercises, simplify each expression.
22. ⓐ \((−7)^3\) ⓑ \(−7^3\)
23. \((7−11)(6−13)\)
- Answer
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16
24. \(63÷(−9)+(−36)÷(−4)\)
25. \(6−3|4(1−2)−(7−5)|\)
- Answer
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\(−12\)
26. \((−2)^4−24÷(13−5)\)
27. \((y+z)^2\) when \(y=−4\) and \(z=7\)
- Answer
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9
28. \(3x^2−2xy+4y^2\) when \(x=−2\) and \(y=−3\)
In the following exercises, translate to an algebraic expression and simplify if possible.
29. the sum of \(−4\) and \(−9\), increased by \(23\)
- Answer
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\((−4+(−9))+23;10\)
30. ⓐ the difference of 17 and −8 ⓑ subtract 17 from −25
Use Integers in Applications
In the following exercise, solve.
31. 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 Palm Springs and the temperature in Whitefield?
- Answer
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\(46°\)
Fractions
Simplify Fractions
In the following exercises, simplify.
32. \(\dfrac{204}{228}\)
33. \(−\dfrac{270x^3}{198y^2}\)
- Answer
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\(−\dfrac{15x^3}{11y^2}\)
Multiply and Divide Fractions
In the following exercises, perform the indicated operation.
34. \(\left(−\dfrac{14}{15}\right)\left(\dfrac{10}{21}\right)\)
35. \(\dfrac{6x}{25}÷\dfrac{9y}{20}\)
- Answer
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\(\dfrac{8x}{15y}\)
36. \(\dfrac{−\frac{4}{9}}{\dfrac{8}{21}}\)
Add and Subtract Fractions
In the following exercises, perform the indicated operation.
37. \(\dfrac{5}{18}+\dfrac{7}{12}\)
- Answer
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\(\dfrac{31}{36}\)
38. \(\dfrac{11}{36}−\dfrac{15}{48}\)
39. ⓐ \(\dfrac{5}{8}+\dfrac{3}{4}\) ⓑ \(\dfrac{5}{8}÷\dfrac{3}{4}\)
- Answer
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ⓐ \(\dfrac{11}{8}\) ⓑ \(\dfrac{5}{6}\)
40. ⓐ \(−\dfrac{3y}{10}−\dfrac{5}{6}\) ⓑ \(−\dfrac{3y}{10}·\dfrac{5}{6}\)
Use the Order of Operations to Simplify Fractions
In the following exercises, simplify.
41. \(\dfrac{4·3−2·5}{−6·3+2·3}\)
- Answer
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\(−\dfrac{1}{6}\)
42. \(\dfrac{4(7−3)−2(4−9)}{−3(4+2)+7(3−6)}\)
43. \(\dfrac{4^3−4^2}{(\dfrac{4}{5})^2}\)
- Answer
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75
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.
44. \(4x^2y^2\) when \(x=\dfrac{2}{3}\) and \(y=−\dfrac{3}{4}\)
45. \(\dfrac{a+b}{a−b}\) when \(a=−4\) and \(b=6\)
- Answer
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\(−15\)
Decimals
Round Decimals
46. Round \(6.738\) to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Add and Subtract Decimals
In the following exercises, perform the indicated operation.
47. \(−23.67+29.84\)
- Answer
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\(6.17\)
48. \(54.3−100\)
49. \(79.38−(−17.598)\)
- Answer
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\(96.978\)
Multiply and Divide Decimals
In the following exercises, perform the indicated operation.
50. \((−2.8)(3.97)\)
51. \((−8.43)(−57.91)\)
- Answer
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488.1813
52. \((53.48)(10)\)
53. \((0.563)(100)\)
- Answer
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\(56.3\)
54. \( \$ 118.35÷2.6\)
55. \(1.84÷(−0.8)\)
- Answer
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\(−23\)
Convert Decimals, Fractions and Percents
In the following exercises, write each decimal as a fraction.
56. \(\dfrac{13}{20}\)
57. \(−\dfrac{240}{25}\)
- Answer
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\(−9.6\)
In the following exercises, convert each fraction to a decimal.
58. \(−\dfrac{5}{8}\)
59. \(\dfrac{14}{11}\)
- Answer
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\(1.\overline{27}\)
In the following exercises, convert each decimal to a percent.
60. \(2.43\)
61. \(0.0475\)
- Answer
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\(4.75 \% \)
Simplify Expressions with Square Roots
In the following exercises, simplify.
62. \(\sqrt{289}\)
63. \(\sqrt{−121}\)
- Answer
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no real number
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercise, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers for each set of numbers
64. \(−8,0,1.95286...,\dfrac{12}{5},\sqrt{36},9\)
Locate Fractions and Decimals on the Number Line
In the following exercises, locate the numbers on a number line.
65. \(\dfrac{3}{4},−\dfrac{3}{4},1\dfrac{1}{3},−1\dfrac{2}{3},\dfrac{7}{2},−\dfrac{5}{2}\)
- Answer
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66. ⓐ \(3.2\) ⓑ \(−1.35\)
Properties of Real Numbers
Use the Commutative and Associative Properties
In the following exercises, simplify.
67. \(\dfrac{5}{8}x+\dfrac{5}{12}y+\dfrac{1}{8}x+\dfrac{7}{12}y\)
- Answer
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\(\dfrac{3}{4}x+y\)
68. \(−32·9·\dfrac{5}{8}\)
69. \(\left(\dfrac{11}{15}+\dfrac{3}{8}\right)+\dfrac{5}{8}\)
- Answer
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\(1\dfrac{11}{15}\)
Use the Properties of Identity, Inverse and Zero
In the following exercises, simplify.
70. \(\dfrac{4}{7}+\dfrac{8}{15}+\left(−\dfrac{4}{7}\right)\)
71. \(\dfrac{13}{15}·\dfrac{9}{17}·\dfrac{15}{13}\)
- Answer
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\(\dfrac{9}{17}\)
72. \(\dfrac{0}{x−3},x\neq 3\)
73. \(\dfrac{5x−7}{0},5x−7\neq 0\)
- Answer
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undefined
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.
74. \(8(a−4)\)
75. \(12\left(\dfrac{2}{3}b+\dfrac{5}{6}\right)\)
- Answer
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\(8b+10\)
76. \(18·\dfrac{5}{6}(2x−5)\)
77. \((x−5)p\)
- Answer
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\(xp−5p\)
78. \(−4(y−3)\)
79. \(12−6(x+3)\)
- Answer
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\(−6x−6\)
80. \(6(3x−4)−(−5)\)
81. \(5(2y+3)−(4y−1)\)
- Answer
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\(y+16\)
Practice Test
1. Find the prime factorization of \(756\).
2. Combine like terms: \(5n+8+2n−1\)
- Answer
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\(7n+7\)
3. Evaluate when \(x=−2\) and \(y=3: \dfrac{|3x−4y|}{6}\)
4. Translate to an algebraic expression and simplify:
ⓐ eleven less than negative eight
ⓑ the difference of \(−8\) and \(−3\), increased by 5
- Answer
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\(−8−11 = −19\)
\((−8−(−3))+5 = 0\)
5. Dushko has nickels and pennies in his pocket. The number of pennies is seven less than four times the number of nickels. Let nn represent the number of nickels. Write an expression for the number of pennies.
6. Round \(28.1458\) to the nearest
ⓐ hundredth ⓑ thousandth
- Answer
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ⓐ \(28.15\) ⓑ \(28.146\)
7. Convert
ⓐ \(\dfrac{5}{11}\) to a decimal ⓑ \(1.15\) to a percent
8. Locate \(\dfrac{3}{5},2.8,and−\dfrac{5}{2}\) on a number line.
- Answer
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In the following exercises, simplify each expression.
9. \(8+3[6−3(5−2)]−4^2\)
10. \(−(4−9)−(9−5)\)
- Answer
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1
11. \(56÷(−8)+(−27)÷(−3)\)
12. \(16−2|3(1−4)−(8−5)|\)
- Answer
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\(−8\)
13. \(−5+2(−3)^2−9\)
14. \(\dfrac{180}{204}\)
- Answer
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\(\dfrac{15}{17}\)
15. \(−\dfrac{7}{18}+\dfrac{5}{12}\)
16. \(\dfrac{4}{5}÷(−\dfrac{12}{25})\)
- Answer
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\(−\dfrac{5}{3}\)
17. \(\dfrac{9−3·9}{15−9}\)
18. \(\dfrac{4(−3+2(3−6))}{3(11−3(2+3))}\)
- Answer
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\(3\)
19. \(\dfrac{5}{13}⋅\dfrac{4}{7}⋅\dfrac{13}{5}\)
20. \(\dfrac{−\dfrac{5}{9}}{\dfrac{10}{21}}\)
- Answer
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\(−\dfrac{7}{6}\)
21. \(−4.8+(−6.7)\)
22. \(34.6−100\)
- Answer
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\(−65.4\)
23. \(−12.04⋅(4.2)\)
24. \(−8÷0.05\)
- Answer
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160
25. \(−\sqrt{121}\)
26. \((\dfrac{8}{13}+\dfrac{5}{7})+\dfrac{2}{7}\)
- Answer
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\(1\dfrac{8}{13}\)
27. \(5x+(−8y)−6x+3y\)
28. ⓐ \(\dfrac{0}{9}\) ⓑ \(\dfrac{11}{0}\)
- Answer
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ⓐ 0 ⓑ undefined
29. \(−3(8x−5)\)
30. \(6(3y−1)−(5y−3)\)
- Answer
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\(13y−3\)