Appendix A — Answer Key, Chapter 3: Square Roots and Perfect Squares

SOL 7.NS.3 · Covers textbook Chapter 3 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 80 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Reference list used throughout: $0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400$.

Lesson 3.1 — Perfect Squares from 0 to 400

Guided practice

1. $36$ unit squares; written with an exponent, $6^2 = 36$.
2. a) $9$ b) $64$ c) $144$
3. a) $225$ b) $400$ c) $0$
4. No. It lies between the consecutive perfect squares $49 = 7^2$ and $64 = 8^2$, and there is no whole number between 7 and 8 to square.
5. $121 = 11^2$ and $169 = 13^2$
6. Between $196 = 14^2$ and $225 = 15^2$

Independent practice

7. a) $49$ b) $121$ c) $196$ d) $289$
8. a) $81$ b) $169$ c) $256$ d) $361$
9. $121$, $144$, $169$, $196$
10. Perfect squares: $25$ ($5^2$), $64$ ($8^2$), $100$ ($10^2$). Not perfect squares: 24, 60, 90.
11. Differences: $3$, $5$, $7$, $9$. The gaps are the odd numbers in order, because each new square adds a row and a column. The next gap is $11$, so the next perfect square is $25 + 11 = 36$, which checks as $6^2 = 36$.
12. Perfect squares only ever end in 0, 1, 4, 5, 6, or 9. Since 7 is not on that list, a number ending in 7 cannot be a perfect square.
13. Between $225 = 15^2$ and $256 = 16^2$
14. Because this standard uses perfect squares from 0 to 400, and $20^2 = 400$ is the last one that fits. The next perfect square, $21^2 = 441$, is greater than 400.
15. $18^2 = 18 \times 18 = 324$ square feet
16. A perfect square comes from multiplying a whole number by itself, and $4 \times 5$ uses two different factors. Any rectangle works for $4 \times 5$, but only a square counts here. 20 lies between $16 = 4^2$ and $25 = 5^2$, so it is not a perfect square.

Exit ticket 3.1

17. $13^2 = 169$ and $16^2 = 256$
18. Yes. $144 = 12^2$, so the whole number is $12$.
19. Between $121 = 11^2$ and $144 = 12^2$
20. A number is a perfect square when that many unit squares can be arranged into a square array with the same number of tiles in every row and every column. For example, 16 tiles form a $4 \times 4$ array, so 16 is a perfect square; 20 tiles cannot form such an array.

Lesson 3.2 — The Positive Square Root

Guided practice

21. $5$
22. $9$
23. $\sqrt{0} = 0$ and $\sqrt{1} = 1$
24. $14$
25. $\sqrt{100} = 10$ centimeters
26. Between $5$ and $6$, since $25 < 30 < 36$.

Independent practice

27. a) $7$ b) $8$ c) $11$ d) $12$
28. a) $13$ b) $15$ c) $16$ d) $20$
29. a) $17$ b) $18$ c) $19$ d) $4$
30. Values: $\sqrt{4} = 2$, $\sqrt{36} = 6$, $\sqrt{81} = 9$, $\sqrt{121} = 11$. Order: $\sqrt{4}$, $\sqrt{36}$, $\sqrt{81}$, $\sqrt{121}$.
31. Between $12$ and $13$, using $144 = 12^2$ and $169 = 13^2$, since $144 < 150 < 169$.
32. Between $14$ and $15$, using $196 = 14^2$ and $225 = 15^2$, since $196 < 200 < 225$.
33. False. $\sqrt{36} = 6$, because $6 \times 6 = 36$. The answer 18 comes from halving, which undoes multiplying by 2, not squaring.
34. $\sqrt{0} = 0$ because $0 \times 0 = 0$, and $\sqrt{1} = 1$ because $1 \times 1 = 1$. Each number is its own square root here.
35. Side: $\sqrt{225} = 15$ feet. Trim all the way around: $4 \times 15 = 60$ feet.
36. The perfect-square neighbors of 170 are $169 = 13^2$ and $196 = 14^2$, and $169 < 170 < 196$. Taking positive square roots keeps the order, so $13 < \sqrt{170} < 14$.

Exit ticket 3.2

37. $\sqrt{64} = 8$ and $\sqrt{289} = 17$
38. $\sqrt{196} = 14$ feet
39. Between $7$ and $8$, since $49 < 55 < 64$.
40. It asks for the positive number that, multiplied by itself, gives the number under the radical sign. In a picture, it asks for the side length of a square whose area is that number.

Lesson 3.3 — How Square Roots and Perfect Squares Undo Each Other

Guided practice

41. $6^2 = 36$ and $\sqrt{36} = 6$
42. $14^2 = 196$ and $\sqrt{196} = 14$
43. $81$. ($\sqrt{81} = 9$, and $9^2 = 81$.)
44. $15$. ($15^2 = 225$, and $\sqrt{225} = 15$.)
45. $\sqrt{64} = 8$ inches. Check: $8 \times 8 = 64$.
46. inverse

Independent practice

47. a) $49$ b) $12$ c) $100$ d) $17$

The last row is identical to the first row: taking the positive square root undoes the squaring and returns the starting number.

49. $\sqrt{289} = 17$ meters. Check: $17 \times 17 = 289$.
50. $\sqrt{169} = 13$ and $13^2 = 169$
51. $\sqrt{400} = 20$ and $20^2 = 400$
52. $s = \sqrt{121} = 11$
53. Squaring and taking the positive square root are inverse operations, so applied one after the other they return the starting number. That makes $\sqrt{9^2} = 9$ directly.
54. Values: $5^2 = 25$, $\sqrt{144} = 12$, $\sqrt{64} = 8$, $3^2 = 9$. Order: $\sqrt{64}$, $3^2$, $\sqrt{144}$, $5^2$ (that is, $8$, $9$, $12$, $25$).
55. Side: $\sqrt{324} = 18$ feet. Edging: $4 \times 18 = 72$ feet.
56. Halving undoes multiplying by 2, not squaring. To undo squaring you take the positive square root, asking what number times itself gives 16. The correct value is $\sqrt{16} = 4$, and the check is $4 \times 4 = 16$. Dana's answer fails its own check, since $8 \times 8 = 64$, not 16.

Exit ticket 3.3

57. $11^2 = 121$ and $\sqrt{121} = 11$
58. $16$
59. $\sqrt{400} = 20$ centimeters
60. Squaring a whole number produces a perfect square, and taking the positive square root of that perfect square brings you back to the whole number you started with. For example, $12^2 = 144$ and $\sqrt{144} = 12$. Because they undo each other, you can always check a square root by squaring your answer.

Chapter 3 Review

Part A — Perfect squares from 0 to 400 (7.NS.3a, 7.NS.3b)

61. a) $4$ b) $100$ c) $225$ d) $324$
62. $225$, $256$, $289$, $324$, $361$
63. Perfect squares: $49$ ($7^2$), $81$ ($9^2$), $196$ ($14^2$). Not perfect squares: 45, 96, 300.
64. Between $256 = 16^2$ and $289 = 17^2$
65. $13^2 = 169$ square centimeters

Part B — Positive square roots (7.NS.3a)

66. a) $2$ b) $6$ c) $10$ d) $16$
67. a) $12$ b) $18$ c) $19$ d) $0$
68. Values: $\sqrt{25} = 5$, $\sqrt{81} = 9$, $\sqrt{196} = 14$, $\sqrt{289} = 17$. Order: $\sqrt{25}$, $\sqrt{81}$, $\sqrt{196}$, $\sqrt{289}$.
69. $\sqrt{144} = 12$ feet
70. Between $9$ and $10$, since $81 < 90 < 100$.
71. Between $19$ and $20$, since $361 < 380 < 400$.

Part C — The inverse relationship (7.NS.3b)

72. $225$
73. $14$
74. $\sqrt{225} = 15$ and $15^2 = 225$
75. $s = \sqrt{256} = 16$
76. Square the answer and see whether the radicand comes back: $13 \times 13 = 169$. Since squaring and taking the positive square root are inverse operations, a correct root always reproduces the radicand.

Part D — Mixed application and reasoning

77. Side: $\sqrt{289} = 17$ feet. Distance around: $4 \times 17 = 68$ feet.
78. First side: $\sqrt{100} = 10$ feet. New side: $\sqrt{400} = 20$ feet. The side length doubled, even though the area became four times as large.
79. The side length is a distance, so it must be positive, and only one positive number squares to 49. Since $7 \times 7 = 49$, the side is $7$ units, and that is the only acceptable answer.
80. The perfect-square neighbors of 260 are $256 = 16^2$ and $289 = 17^2$, and $256 < 260 < 289$. Since 260 sits strictly between two consecutive perfect squares, no whole number squares to it, so $\sqrt{260}$ is not a whole number. It lies between $16$ and $17$.

Workbook-only items

Page 2, fill in the blanks. The raised 2 is an exponent. $4^2$ means $4 \times$ 4, which equals 16. $4^2$ does not mean $4 \times$ 2. A square with a side of 4 units has an area of 16 square units.

Page 2, complete the rule. area $=$ (side length)$^2$

Page 2, count and record. 4 rows, 4 tiles in each row, 16 total tiles, written $4^2$.

Page 3, growing-squares table. Side 1 → 1; side 2 → 4; side 3 → 9; side 4 → 16; side 5 → 25; side 6 → 36.

Page 3, gaps. $1 \to 4$: 3. $4 \to 9$: 5. $9 \to 16$: 7. $16 \to 25$: 9. The gaps are the odd numbers in order.

Page 4, full table. $n = 0$ through $20$ gives $n^2 = 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400$.

Page 8, label the parts. $\sqrt{\phantom{x}}$ is the radical sign. 49 is the radicand. 7 is the positive square root of 49.

Page 8, fill in the blanks. $\sqrt{49} = 7$ because 7 $\times$ 7 $= 49$. A square root answers: here is the area of a square, how long is one side? We take the positive root because a side length is a distance, and distances are never negative.

Page 8, watch out. $\sqrt{36} =$ 6 because 6 $\times$ 6 $= 36$.

Page 10, follow the reasoning. $5 < \sqrt{30} < 6$

Page 12, fill in the blanks. Squaring takes a side length and gives back an area. Taking the positive square root takes an area and gives back a side length. Operations that undo each other are called inverse operations. $\sqrt{n^2} = n$ and $\left(\sqrt{n^2}\right)^2 = n^2$.

Page 13, checks. Item 45 check: $8 \times 8 = 64$. Item 49 check: $17 \times 17 = 289$.

Page 14, item 54 values. $8$, $9$, $12$, $25$.

Page 14, item 56 check. $\sqrt{16} = 4$; check $4 \times 4 = 16$.