Appendix A — Answer Key, Chapter 8: Simplifying Numerical Expressions

SOL 7.PFA.2 (a) · Covers textbook Chapter 8 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 103 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Multi-step answers show the intermediate lines on purpose. The steps are what is being taught; a bare final value hides whether the order of operations was followed.

Tier reference used throughout: 1 grouping symbols (innermost first) · 2 exponents and perfect-square roots · 3 multiplication and division, left to right · 4 addition and subtraction, left to right.

Lesson 8.1 — The Order of Operations

Guided practice

1. $9 + 6 \times 2 = 9 + 12 = 21$. The first tier used is tier 3, multiplication.
2. $(9 + 6) \times 2 = 15 \times 2 = 30$. The parentheses moved the addition ahead of the multiplication, which changed the value from 21 to 30.
3. $20 - 12 \div 4 = 20 - 3 = 17$
4. $36 \div 9 \times 2 = 4 \times 2 = 8$. The leftmost operation is $36 \div 9$.
5. $5 \times (11 - 4) + 8 = 5 \times 7 + 8 = 35 + 8 = 43$
6. $48 \div (6 + 2) - 3 = 48 \div 8 - 3 = 6 - 3 = 3$

Independent practice

7. a) $8 + 15 = 23$ b) $13 \times 3 = 39$ c) $40 + 3 = 43$ d) $8 \times 8 = 64$
8. a) $40 - 24 = 16$ b) $34 \times 4 = 136$ c) $5 + 2 = 7$ d) $40 \div 10 = 4$
9. $60 \div 5 \times 3 = 12 \times 3 = 36$
10. $18 - 4 + 7 = 14 + 7 = 21$
11. $7 \times 4 - 15 \div 5 = 28 - 3 = 25$
12. $(25 - 9) \div 4 + 6 \times 2 = 16 \div 4 + 6 \times 2 = 4 + 12 = 16$
13. $100 - (36 \div 4 + 11) \times 3 = 100 - (9 + 11) \times 3 = 100 - 20 \times 3 = 100 - 60 = 40$
14. $4 \times 3 + 2 \times 5 = 12 + 10 = 22$, so the total cost is $22.
15. The value is $15$. Addition and subtraction share one tier, so they are done left to right: $18 - 5 = 13$ first, then $13 + 2 = 15$. Adding $5 + 2$ first would treat addition as outranking subtraction, which it does not.
16. Riley added before multiplying. Multiplication is tier 3 and addition is tier 4, so the multiplication comes first: $6 + 2 \times 9 = 6 + 18 = 24$.

Exit ticket 8.1

17. $12 + 8 \div 2 = 12 + 4 = 16$
18. $(12 + 8) \div 2 = 20 \div 2 = 10$
19. $9 \times 3 - 20 \div 4 = 27 - 5 = 22$
20. Without an agreed order, the same expression could be simplified two ways and give two different values, so nobody could rely on written arithmetic. The order of operations is a shared convention that makes every expression mean exactly one thing.

Lesson 8.2 — Exponents and Perfect-Square Roots Inside Expressions

Guided practice

21. a) $2^3 = 8$ b) $3^4 = 81$ c) $10^2 = 100$ d) $7^1 = 7$
22. $4 + 5^2 = 4 + 25 = 29$
23. $6 \times 2^3 = 6 \times 8 = 48$
24. $\sqrt{64} + 12 = 8 + 12 = 20$
25. $3^2 + \sqrt{16} \times 2 = 9 + 4 \times 2 = 9 + 8 = 17$
26. $(2 + 3)^2 - 15 = 5^2 - 15 = 25 - 15 = 10$. Simplify inside the parentheses first, because grouping symbols are tier 1 and exponents are tier 2.

Independent practice

27. a) $4$ b) $16$ c) $125$ d) $1$
28. a) $6$ b) $10$ c) $12$ d) $15$
29. $50 - 4^2 \times 2 = 50 - 16 \times 2 = 50 - 32 = 18$
30. $\sqrt{81} \times 3 + 2^3 = 9 \times 3 + 8 = 27 + 8 = 35$
31. $(6 - 2)^3 \div 8 = 4^3 \div 8 = 64 \div 8 = 8$
32. $\sqrt{9 + 16} + 5^2 = \sqrt{25} + 25 = 5 + 25 = 30$
33. $3^3 - \sqrt{49} \times 2 + 1 = 27 - 7 \times 2 + 1 = 27 - 14 + 1 = 13 + 1 = 14$
34. $100 \div (2 + 3)^2 \times 3 = 100 \div 5^2 \times 3 = 100 \div 25 \times 3 = 4 \times 3 = 12$
35. $4 \times \sqrt{144} = 4 \times 12 = 48$ feet. The side length is $\sqrt{144} = 12$ feet, and a square has four equal sides.
36. $2^3 = 2 \times 2 \times 2 = 8$ and $3^2 = 3 \times 3 = 9$, so they are not equal. The base says which number is repeated and the exponent says how many times, so swapping them describes a different product.
37. Tomas added before applying the exponent. Exponents are tier 2 and addition is tier 4, so the power comes first: $3 + 2^3 = 3 + 8 = 11$.

Exit ticket 8.2

38. $7 + 2^4 = 7 + 16 = 23$
39. $\sqrt{100} - 3^2 = 10 - 9 = 1$
40. $(4 + 5)^2 \div 3 = 9^2 \div 3 = 81 \div 3 = 27$
41. The radical sign is a grouping symbol, so everything under it is simplified first: $\sqrt{9 + 16} = \sqrt{25} = 5$. Taking the roots separately gives $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$, a different expression and a different value.

Lesson 8.3 — Brackets and Absolute Value Bars

Guided practice

42. $2[9 + (4 - 1)] = 2[9 + 3] = 2[12] = 24$. Simplify the innermost group, $(4 - 1)$, first.
43. $[30 - (6 \times 4)] + 5 = [30 - 24] + 5 = 6 + 5 = 11$
44. $|-9| = 9$
45. $|5 - 12| = |-7| = 7$
46. $|2 - 8| + 3^2 = |-6| + 9 = 6 + 9 = 15$
47. $4[|7 - 7| + 5] = 4[|0| + 5] = 4[0 + 5] = 4[5] = 20$

Independent practice

48. a) $15$ b) $20$ c) $0$ d) $|-4| + |4| = 4 + 4 = 8$
49. $3[14 - (2 + 5)] = 3[14 - 7] = 3[7] = 21$
50. $[45 \div (2 + 7)] \times 4 = [45 \div 9] \times 4 = 5 \times 4 = 20$
51. $50 - 3|4 - 9| = 50 - 3|-5| = 50 - 3 \times 5 = 50 - 15 = 35$
52. $[2^3 + |1 - 5|] \div 3 = [8 + |-4|] \div 3 = [8 + 4] \div 3 = 12 \div 3 = 4$
53. $\sqrt{|9 - 25|} = \sqrt{|-16|} = \sqrt{16} = 4$
54. $2[(3 + 2)^2 - |10 - 14|] = 2[5^2 - |-4|] = 2[25 - 4] = 2[21] = 42$
55. $[100 - 4^2] \div |0 - 7| = [100 - 16] \div |-7| = 84 \div 7 = 12$
56. $|9 - (-5)| = |14| = 14$ degrees. (The expression $|-5 - 9| = |-14| = 14$ is equally correct, since the bars report the distance between the two temperatures either way.)
57. Both equal $7$. $|3 - 10| = |-7| = 7$ and $|10 - 3| = |7| = 7$. The bars report the distance between 3 and 10 on the number line, and a distance does not depend on which endpoint you start from.
58. Priya dropped the bars before simplifying inside them. The bars are a grouping symbol: finish inside, then take the distance from zero. $|4 - 9| + 6 = |-5| + 6 = 5 + 6 = 11$.

Exit ticket 8.3

59. $|8 - 15| = |-7| = 7$
60. $4[6 - (1 + 3)] = 4[6 - 4] = 4[2] = 8$
61. $[3^2 + |2 - 6|] \div 13 = [9 + |-4|] \div 13 = [9 + 4] \div 13 = 13 \div 13 = 1$
62. The bars hold $4 - 9$ together as one package, exactly the way parentheses would, so nothing outside them can be done until the inside is a single number. Finish the subtraction first, getting $-5$, and only then take the distance from zero, which is $5$.

Lesson 8.4 — Properties of Real Numbers as Shortcuts

Guided practice

63. Commutative property of addition
64. Associative property of multiplication
65. $28 + 15 + 2 = (28 + 2) + 15 = 30 + 15 = 45$. Pair 28 and 2 because they make 30, a friendly number.
66. $5 \times 17 \times 2 = (5 \times 2) \times 17 = 10 \times 17 = 170$
67. $7(20 + 3) = 7(20) + 7(3) = 140 + 21 = 161$
68. Multiplicative identity property

Independent practice

69. a) Additive identity property b) Multiplicative inverse property c) Distributive property d) Commutative property of multiplication
70. $47 + 68 + 53 = (47 + 53) + 68 = 100 + 68 = 168$
71. $4 \times 19 \times 25 = (4 \times 25) \times 19 = 100 \times 19 = 1{,}900$
72. $6(50 + 7) = 6(50) + 6(7) = 300 + 42 = 342$. Check: $6(57) = 342$.
73. $9(30 - 2) = 9(30) - 9(2) = 270 - 18 = 252$. Check: $9(28) = 252$.
74. $58 + 27 + (-58) = 58 + (-58) + 27 = 0 + 27 = 27$. Properties used: commutative property of addition, additive inverse property, additive identity property.
75. $\left(\tfrac{1}{4} \times 4\right) \times 23 = 1 \times 23 = 23$. Properties used: multiplicative inverse property, multiplicative identity property.
76. $5(20 + 4) - 3^2 = 5(20) + 5(4) - 3^2 = 100 + 20 - 9 = 120 - 9 = 111$. Check: $5(24) - 9 = 120 - 9 = 111$.
77. $8(12) + 8(3) = 96 + 24 = 120$, and as a single product $8(12 + 3) = 8(15) = 120$. The total is $120. The two forms match because of the distributive property.
78. $10 - 4 = 6$ but $4 - 10 = -6$, so swapping the two numbers changes the value and subtraction is not commutative. The properties are shortcuts rather than exceptions because each one is a statement that two expressions have the same value. You are allowed to take the easier road only because it arrives at the identical number the strict order would give.
79. Devon multiplied the first term only. The factor outside must multiply every term inside: $3(8 + 5) = 3(8) + 3(5) = 24 + 15 = 39$. Check: $3(13) = 39$.

Exit ticket 8.4

80. Associative property of addition
81. $25 \times 13 \times 4 = (25 \times 4) \times 13 = 100 \times 13 = 1{,}300$
82. $6(40 + 3) = 6(40) + 6(3) = 240 + 18 = 258$. Check: $6(43) = 258$.
83. Each property is a statement that two ways of writing the same computation are equal, so applying one rewrites the expression without changing what it is worth. A shortcut only reorders or regroups within one operation; it never moves a number across tiers of the order of operations.

Chapter 8 Review

Part A — Order of operations (7.PFA.2a)

84. $15 + 6 \times 3 = 15 + 18 = 33$
85. $(15 + 6) \times 3 = 21 \times 3 = 63$
86. $72 \div 9 \times 2 = 8 \times 2 = 16$
87. $40 - 24 \div 6 + 5 = 40 - 4 + 5 = 36 + 5 = 41$
88. $9 \times (12 - 5) - 18 \div 3 = 9 \times 7 - 6 = 63 - 6 = 57$

Part B — Exponents and perfect-square roots (7.PFA.2a)

89. a) $27$ b) $16$ c) $25$ d) $1{,}000$
90. a) $4$ b) $11$ c) $14$ d) $20$
91. $4^2 + \sqrt{81} \times 2 = 16 + 9 \times 2 = 16 + 18 = 34$
92. $(7 - 3)^3 \div \sqrt{64} = 4^3 \div \sqrt{64} = 64 \div 8 = 8$

Part C — Brackets and absolute value bars (7.PFA.2a)

93. $5[13 - (4 + 6)] = 5[13 - 10] = 5[3] = 15$
94. $|7 - 16| + 2^3 = |-9| + 8 = 9 + 8 = 17$
95. $[6^2 - |3 - 11|] \div 7 = [36 - |-8|] \div 7 = [36 - 8] \div 7 = 28 \div 7 = 4$
96. $2[|5 - 12| + \sqrt{100}] = 2[|-7| + 10] = 2[7 + 10] = 2[17] = 34$

Part D — Properties of real numbers (7.PFA.2a)

97. a) Associative property of addition b) Multiplicative identity property c) Distributive property
98. $36 + 89 + 64 = (36 + 64) + 89 = 100 + 89 = 189$
99. $8(60 + 4) = 8(60) + 8(4) = 480 + 32 = 512$. Check: $8(64) = 512$.

Part E — Mixed application and reasoning

100. $3[\sqrt{49} + (2 + 6)] - 4^2 = 3[7 + 8] - 4^2 = 3[15] - 16 = 45 - 16 = 29$
101. $4 \times \sqrt{225} - 6 = 4 \times 15 - 6 = 60 - 6 = 54$ feet of fencing. The side length is $\sqrt{225} = 15$ feet, the full perimeter is 60 feet, and the gate removes 6 feet.
102. $|2 - 9| \times 3 = |-7| \times 3 = 7 \times 3 = 21$, while $2 - 9 \times 3 = 2 - 27 = -25$. In the first expression the bars group $2 - 9$, so the subtraction happens first and the result becomes a distance of 7. In the second there are no grouping symbols, so $9 \times 3$ is done first as tier 3 and the subtraction happens last. Grouping symbols changed which operation came first, and the absolute value also removed the negative.
103. Sam subtracted before clearing the grouping symbols, and grouping symbols are tier 1. The correct work is

$$\begin{aligned} 20 - 2[3 + (4 \times 2)] &= 20 - 2[3 + 8] \\ &= 20 - 2[11] \\ &= 20 - 22 \\ &= -2 \end{aligned}$$

The correct value is $-2$.

Workbook-only items

Page 2, two students. Student A gets $10 \times 4 = 40$. Student B gets $7 + 12 = 19$. Student B is correct, so $7 + 3 \times 4 = 19$.

Page 2, fill in the blanks. Tier 1 is grouping symbols; simplify the innermost group first. Tier 2 is exponents and perfect-square roots. Tier 3 is multiplication and division, worked left to right. Tier 4 is addition and subtraction, worked left to right.

Page 2, watch the traps. $24 \div 6 \times 2 = 8$. $18 - 5 + 2 = 15$.

Page 7, power table.

Page 7, fill in the blanks. In $2^3$ the base is 2 and the exponent is 3, so $2^3 = 2 \times 2 \times 2 = 8$. $2^3$ is NOT $2 \times 3$. $2^3 = 8$ but $3^2 = 9$. The radical sign is also a grouping symbol, so $\sqrt{25 + 11} = \sqrt{36} = 6$.

Page 11, fill in the blanks. Brackets do exactly what parentheses do; they are used so the layers are easy to see. With nested groups, simplify the innermost group first. In $2[8]$ the 2 means multiplication, so $2[8] = 16$. This book never uses braces.

Page 15, property table. $6 + 9 = 9 + 6$ · $2 \times 11 = 11 \times 2$ · $(7 + 2) + 8 = 7 + (2 + 8)$ · $(3 \times 5) \times 2 = 3 \times (5 \times 2)$ · $6(10 + 4) = 6(10) + 6(4)$ · $8 + 0 = 8$ · $14 \times 1 = 14$ · $46 + (-46) = 0$ · $9 \times \tfrac{1}{9} = 1$

Page 15, boundary line. $10 - 4 = 6$ but $4 - 10 = -6$, so subtraction is not commutative.