Algebra 1 Workbook — Chapter 11: Radical Expressions

SOL A.EO.4 (a, b, c, d) · Companion to Textbook Chapter 11

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 120.

PAGE 1 — Chapter opener

Chapter 11 · Radical Expressions

Standard: A.EO.4 (a, b, c, d)

In this chapter you will:

• Put square roots of whole numbers into simplest form
• Locate irrational square roots between consecutive integers
• Simplify cube roots of integers, including negatives
• Add, subtract, and multiply numeric radicals
• Show that roots split products, never sums
• Justify $a^{1/2} = \sqrt{a}$ and $a^{1/3} = \sqrt[3]{a}$ from the product law

Words to know: square root · principal square root · perfect square · radicand · simplest form · perfect-square factor · cube root · perfect cube · like radicals · product property · rational exponent

Convention: $\sqrt{a}$ is the non-negative number whose square is $a$. So $\sqrt{49} = 7$, never $\pm 7$.

Convention: simplest form looks only under the bar. $2\sqrt{18}$ fails because $18$ still hides a $9$.

Convention: rational exponents in this chapter are only $\tfrac12$ and $\tfrac13$. No dividing radicals, no rationalizing denominators.

Calculator. Algebra 1 has no no-calculator standards. Use one to confirm: $\sqrt{72}$ and $6\sqrt{2}$ must show the same decimal.

PAGE 2 — Perfect squares

11.1 Square Roots and Simplest Form

FIGURE: fig1-perfect-square-reference-table.png (full width)

Fill in the blanks.

A square root asks for the ____________________ number whose square is under the bar.

A whole number is a perfect square when it appears in the ____________________ row of the figure.

1. List $n^2$ for $n = 1$ through $15$ from the figure: _______________________

   A whole number is a perfect square when it appears in which row? ______

2. $\sqrt{81} =$ ______ $\sqrt{169} =$ ______

   Why neither answer uses $\pm$: _______________________

PAGE 3 — Largest perfect-square factor

Pull Out the Largest Square

FIGURE: fig2-largest-perfect-square-factor.png (full width)

Complete the frame.

$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ whenever $a \ge$ ______ and $b \ge$ ______.

3. Factor pairs of $72$: _______________________

   Perfect squares among them: ______ Largest: ______ Simplest form: ______

4. If you pull out $4$ instead of $36$: unfinished form ______ Still hiding: ______

PAGE 4 — Simplest-form checklist

The Test for Simplest Form

FIGURE: fig4-simplest-form-checklist.png (full width)

The one question: Does the radicand still contain a perfect-square factor other than $1$?

5. $\sqrt{15}$: pass / fail ______ $\sqrt{20}$: ______ → ______ $2\sqrt{18}$: ______ → ______

6. Explain. Why the coefficient in $5\sqrt{7}$ does not decide simplest form.

   ---

PAGE 5 — Practice · simplify square roots

Practice · Simplest Form

7. Evaluate.

8. Simplify.

9. Simplify.

10. $\sqrt{48} =$ ______ $\sqrt{200} =$ ______

PAGE 6 — Practice · reasoning and errors

Practice · Why Largest Matters

11. Explain. $\sqrt{72} = 6\sqrt{2}$ versus unfinished $2\sqrt{18}$.

    ---

12. Apply it. Patio diagonal $\sqrt{128}$ ft → simplest form: ______

13. Error analysis. Student: $\sqrt{50} = 25\sqrt{2}$. Error: _______________________ Correct: ______

14. Error analysis. Student: $2\sqrt{18}$ is simplest "because of the $2$." Error: _______________________ Finished: ______

15. Simplest form? If not, finish.

    a) $\sqrt{14}$ ______ b) $\sqrt{45}$ ______ c) $7\sqrt{3}$ ______ d) $\sqrt{72}$ ______

16. a) $\sqrt{80} = \sqrt{\underline{\hspace{2cm}} \cdot 5} = \underline{\hspace{2cm}}\sqrt{5}$

    b) $\sqrt{112} = \sqrt{\underline{\hspace{2cm}} \cdot 7} = \underline{\hspace{2cm}}\sqrt{7}$

PAGE 7 — Exit ticket 11.1

Exit Ticket · 11.1

17. $\sqrt{64} =$ ______ $\sqrt{121} =$ ______

18. $\sqrt{72} =$ ______ $3\sqrt{20} =$ ______

19. $\sqrt{15}$ simplest? ______ $2\sqrt{18}$ simplest? ______ Finish failures: ______

20. Explain. What $\sqrt{49}$ means, and why the answer is $7$ not $\pm 7$.

    ---

PAGE 8 — Number line

11.2 Estimating Square Roots

FIGURE: fig3-locating-radicals-on-a-number-line.png (full width)

Frame. Sandwich the radicand between consecutive perfect squares.

21. $\sqrt{2}$: inequality ______ decimal ______

    $\sqrt{10}$: inequality ______ decimal ______

22. $\sqrt{45} =$ ______ $\approx$ ______ between ______ and ______

    $\sqrt{72} =$ ______ $\approx$ ______ between ______ and ______

23. Explain. Why $\sqrt{45}$ and $3\sqrt{5}$ mark the same point.

    ---

24. $\sqrt{30}$ lies between ______ and ______ Sandwich: ______

25. $\sqrt{75} =$ ______ lies between ______ and ______

26. Larger: $\sqrt{80}$ or $9$? ______ Why: _______________________ Decimal check: ______

PAGE 9 — Practice · estimate

Practice · Between Integers

27. Name the two consecutive integers.

28. Simplify, then estimate to the nearest thousandth.

29. Order least to greatest: $\sqrt{10}$, $3$, $\sqrt{8}$, $2\sqrt{2}$.

30. Apply it. Wire $\sqrt{162}$ ft → simplest ______ between ______ and ______ $\approx$ ______

PAGE 10 — Practice · compare and place

Practice · Compare Without a Calculator

31. Explain. How the perfect-square table places $\sqrt{40}$.

    ---

32. Error analysis. Student: $\sqrt{50}$ between $5$ and $6$. Error: _______________________ Correct pair: ______

33. $5\sqrt{2}$ ? $7$ (greater / less / equal) ______ Show with a perfect square: _______________________

34. $\sqrt{128} =$ ______ between ______ and ______

35. Square of area $52$ → side ______ $\approx$ ______ m (nearest tenth)

36. a) $8 < \sqrt{\underline{\hspace{2cm}}} < 9$ with simplest form $6\sqrt{2}$ Radicand: ______

    b) $4\sqrt{3}$ lies between ______ and ______

PAGE 11 — Exit ticket 11.2

Exit Ticket · 11.2

37. $\sqrt{60}$ lies between ______ and ______

38. $\sqrt{45} =$ ______ $\approx$ ______

39. Order: $\sqrt{20}$, $4$, $2\sqrt{5}$ → _______________________

40. Explain. Why $\sqrt{72}$ and $6\sqrt{2}$ mark the same point on the figure.

    ---

PAGE 12 — Perfect cubes

11.3 Cube Roots

FIGURE: fig5-perfect-cube-reference-table.png (full width)

Fill in the blanks.

Every integer has exactly ______ real cube root, and it carries the ____________________ of the integer.

41. From the figure, $n^3$ for $n = 1$ to $10$: _______________________

    $(-n)^3$: _______________________ Boxed claim in your words: _______________________

42. $\sqrt[3]{64} =$ ______ $\sqrt[3]{-64} =$ ______

    Why a cube root can be negative: _______________________

PAGE 13 — Largest perfect-cube factor

Pull Out the Largest Cube

FIGURE: fig6-largest-perfect-cube-factor.png (full width)

Complete the frame.

$\sqrt[3]{a}\sqrt[3]{b} = \sqrt[3]{ab}$ for ____________________ integers $a$ and $b$.

43. Factor pairs of $54$: _______________________ Perfect cubes: ______ Finished: ______

44. Why $9$ does not count: _______________________

45. $\sqrt[3]{24} =$ ______ $\sqrt[3]{-16} =$ ______

46. Explain. Why cube roots need no sign restriction on the product property.

    ---

PAGE 14 — Practice · cube roots

Practice · Simplify Cube Roots

47. Evaluate.

48. Simplify.

49. Simplify.

50. $5\sqrt[3]{16} =$ ______ $-2\sqrt[3]{54} =$ ______

PAGE 15 — Practice · cube-root reasoning

Practice · Signs and Errors

51. Explain. $\sqrt{-8}$ versus $\sqrt[3]{-8}$.

    ---

52. Apply it. Cube volume $250$ ft$^3$ → edge ______

53. Error analysis. Student: $\sqrt[3]{-54} = 3\sqrt[3]{2}$. Sign error: _______________________ Correct: ______

54. Error analysis. Student: $\sqrt[3]{54} = \sqrt[3]{9 \cdot 6} = 3\sqrt[3]{6}$. Error: _______________________ Correct: ______

55. Simplest? If not, finish.

    a) $\sqrt[3]{7}$ ______ b) $\sqrt[3]{40}$ ______ c) $2\sqrt[3]{9}$ ______ d) $\sqrt[3]{-128}$ ______

56. a) $\sqrt[3]{80} = \sqrt[3]{\underline{\hspace{2cm}} \cdot 10} = \underline{\hspace{2cm}}\sqrt[3]{10}$

    b) $\sqrt[3]{-162} = \sqrt[3]{\underline{\hspace{2cm}} \cdot 6} = \underline{\hspace{2cm}}\sqrt[3]{6}$

PAGE 16 — Exit ticket 11.3

Exit Ticket · 11.3

57. $\sqrt[3]{125} =$ ______ $\sqrt[3]{-64} =$ ______

58. $\sqrt[3]{40} =$ ______ $\sqrt[3]{-54} =$ ______

59. $\sqrt[3]{16}$ simplest? ______ If not: ______

60. Explain. Why every integer has exactly one real cube root, carrying the integer's sign.

    ---

PAGE 17 — Sorting like radicals

11.4 Adding, Subtracting, and Multiplying Radicals

FIGURE: fig7-sorting-like-radicals.png (full width)

Frame. Simplify first — the piles are invisible until you do.

61. Seven terms simplified: _______________________

    Three totals: ______ ______ ______ Finished sum: ______

62. Explain. Why $\sqrt{2}$ and $\sqrt[3]{2}$ cannot combine.

    ---

PAGE 18 — Roots never split sums

Products Hold · Sums Do Not

FIGURE: fig8-root-of-a-sum-counterexample.png (full width)

63. $\sqrt{9 + 16} =$ ______ $\sqrt{9} + \sqrt{16} =$ ______ Conclusion: _______________________

64. Subtraction counterexample: $\sqrt{25 - 9} =$ ______ $\sqrt{25} - \sqrt{9} =$ ______

    Shaded failing rows: _______________________

65. $5\sqrt{7} - 2\sqrt{7} + \sqrt{7} =$ ______

66. $\sqrt{18} + \sqrt{50} - \sqrt{8} =$ ______

67. $(2\sqrt{5})(3\sqrt{20}) =$ ______

68. $\sqrt[3]{16} + 5\sqrt[3]{2} =$ ______

PAGE 19 — Practice · combine and multiply

Practice · Like Radicals and Products

69. Simplify.

70. Simplify.

71. Simplify.

72. Simplify.

PAGE 20 — Practice · mixed radicals

Practice · Reasoning with Radicals

73. Seven-term sum, finished without looking at pile totals: ______ Check: ______

74. Explain. Why $\sqrt{12} + \sqrt{3} = 3\sqrt{3}$, but $\sqrt{12} + \sqrt{2}$ stays two terms.

    ---

75. Apply it. Panels $\sqrt{48}$ and $\sqrt{75}$ ft → simplified ______ and ______ Difference: ______

76. Error analysis. Student: $\sqrt{9 + 16} = 7$. False rule: _______________________ Correct value: ______

77. Error analysis. Student: $\sqrt{18} + \sqrt{8} = \sqrt{26}$. Error: _______________________ Correct: ______

78. $2\sqrt{50} + 3\sqrt{32} - \sqrt{18} =$ ______

79. $(6\sqrt{3})(\sqrt{12}) - \sqrt{27} =$ ______

80. a) $5\sqrt{2} + \underline{\hspace{2cm}}\sqrt{2} = 12\sqrt{2}$

    b) $\sqrt{a} \cdot \sqrt{b} = \sqrt{\underline{\hspace{2cm}}}$ for $a \ge 0$, $b \ge 0$

PAGE 21 — Exit ticket 11.4

Exit Ticket · 11.4

81. $3\sqrt{5} + 8\sqrt{5} - \sqrt{5} =$ ______

82. $\sqrt{50} + \sqrt{32} =$ ______

83. $(2\sqrt{6})(3\sqrt{6}) =$ ______

84. Counterexample for $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$: _______________________

    Product property that does hold: _______________________

PAGE 22 — Rational-exponent bridge

11.5 Rational Exponents

FIGURE: fig9-rational-exponent-bridge.png (full width)

Fill in the blanks.

$a^{1/2} \cdot a^{1/2} = a^{\underline{\hspace{2cm}}} = a$, so $a^{1/2} = \sqrt{a}$ for $a \ge$ ______.

Only rational exponents in this chapter: ______ and ______.

85. Write the product-law argument forcing $a^{1/2} = \sqrt{a}$.

    ---

    Boxed conclusion: _______________________

86. Three-factor argument for $a^{1/3}$: _______________________

    Holds for: _______________________

87. $\sqrt{5} =$ ______ Check: ______

    $\sqrt[3]{-8} =$ ______ Check: ______

PAGE 23 — Decimal checks

Agreement Is Evidence · Disagreement Is Proof

FIGURE: fig10-decimal-check-of-a-simplification.png (full width)

88. Claim $\sqrt{45} = 3\sqrt{5}$: left ______ right ______ verdict ______

    Claim $\sqrt{45} = 9\sqrt{5}$: left ______ right ______ verdict ______

89. Explain. Why disagreement proves an error, while agreement is only strong evidence.

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PAGE 24 — Blank practice tables

Fill the Factor Tables

FIGURE: fig11-blank-radical-practice-tables.png (full page)

90. Fill every cell of Table A and Table B.

Table A reminders: largest perfect-square factor · split as $\sqrt{\text{square} \cdot \text{rest}}$ · coefficient times $\sqrt{\text{rest}}$.

Table B reminders: largest perfect-cube factor · sign rides with the cube · $9$ is a square, not a cube.

PAGE 25 — Practice · rewrite both ways

Practice · Radical $\leftrightarrow$ Exponent

91. Rewrite in exponent form.

92. Rewrite in radical form; evaluate if an integer.

93. Rewrite and evaluate.

94. Show $16^{1/2} \cdot 16^{1/2} = 16$. Forces $16^{1/2} =$ ______

95. Show three factors of $27^{1/3}$ multiply to $27$. Value: ______

96. Decimal check · verdict.

PAGE 26 — Practice · justify and apply

Practice · Forced Meanings

97. Explain. Why $a^{1/2} = \sqrt{a}$ is forced, not chosen.

    ---

98. Apply it. Square garden area $50$ m$^2$.

    Radical side: ______ Exponent form: ______ Decimal (hundredth): ______

99. Error analysis. Student: $9^{1/2} = \tfrac19$. Error: _______________________ Correct: ______

100. Error analysis. Student: $(-8)^{1/2} = -4$. Two problems: _______________________

101. a) $\sqrt{80} = \underline{\hspace{2cm}}^{1/2} = \underline{\hspace{2cm}}\sqrt{5}$

     b) $\sqrt[3]{-54} = (-54)^{1/3} = \underline{\hspace{2cm}}$

102. $\sqrt[3]{-135}$: largest cube ______ split ______ simplest ______

PAGE 27 — Exit ticket 11.5

Exit Ticket · 11.5

103. $\sqrt{121} =$ ______ $^{1/2} =$ ______ $\sqrt[3]{-8} =$ ______ $^{1/3} =$ ______

104. Product-law argument for $a^{1/2} = \sqrt{a}$:

     ---

105. Decimal check: $\sqrt{45} = 5\sqrt{3}$ → verdict ______ Correct: ______

106. $27^{1/3} =$ ______ $=$ ______

PAGE 28 — Chapter 11 review · Part A

Chapter 11 Review

Part A · Square roots

FIGURE: fig1-perfect-square-reference-table.png (half width)

107. $\sqrt{81} =$ ______ $\sqrt{196} =$ ______ Four larger squares from the figure: _______________________

FIGURE: fig2-largest-perfect-square-factor.png (full width)

108. Largest square in $72$: ______ Steps to $6\sqrt{2}$: _______________________

FIGURE: fig4-simplest-form-checklist.png (full width)

109. $2\sqrt{18}$ fails because ______ → ______ $\sqrt{20}$ fails because ______ → ______

110. a) $\sqrt{48} =$ ______ b) $5\sqrt{8} =$ ______ c) $\sqrt{200} =$ ______ d) $-3\sqrt{75} =$ ______

111. $\sqrt{40}$ between ______ and ______ Simplest ______ $\approx$ ______

PAGE 29 — Chapter 11 review · Parts B and C

Chapter 11 Review (continued)

Part B · Cube roots

FIGURE: fig5-perfect-cube-reference-table.png (half width)

112. $\sqrt[3]{125} =$ ______ $\sqrt[3]{-1000} =$ ______ Why one real cube root: _______________________

FIGURE: fig6-largest-perfect-cube-factor.png (full width)

113. $\sqrt[3]{-54} =$ ______ Why $9$ is ineligible: _______________________

114. a) $\sqrt[3]{24} =$ ______ b) $\sqrt[3]{-16} =$ ______ c) $2\sqrt[3]{128} =$ ______ d) $\sqrt[3]{-135} =$ ______

Part C · Operations

FIGURE: fig7-sorting-like-radicals.png (full width)

115. Finished seven-term sum: ______

116. a) $\sqrt{50} + \sqrt{18} - \sqrt{8} =$ ______

b) $\sqrt{12} \cdot \sqrt{27} =$ ______

c) $(3\sqrt{5})(2\sqrt{20}) =$ ______

d) $\sqrt[3]{16} + 4\sqrt[3]{2} =$ ______

FIGURE: fig8-root-of-a-sum-counterexample.png (full width)

117. Show $\sqrt{9 + 16} \neq \sqrt{9} + \sqrt{16}$: ______ Product property that holds: ______

PAGE 30 — Chapter 11 review · Part D

Chapter 11 Review (continued)

Part D · Rational exponents

FIGURE: fig9-rational-exponent-bridge.png (full width)

118. Justify $a^{1/2} = \sqrt{a}$: _______________________

     $49^{1/2} =$ ______ $=$ ______ $(-8)^{1/3} =$ ______ $=$ ______

FIGURE: fig10-decimal-check-of-a-simplification.png (full width)

119. $\sqrt{45} = 3\sqrt{5}$ verdict ______ $\sqrt{45} = 5\sqrt{3}$ verdict ______

     Two mistakes caught: _______________________

120. Apply it. Plaza area $72$ m$^2$.

     Side (radical): ______ Exponent form: ______ Decimal (hundredth): ______

     Perimeter: ______

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Radical bars and indices. Set every radical bar long enough to cover the whole radicand, and set cube-root indices at no smaller than 70% of body size. $\sqrt[3]{-54}$ must not lose its minus sign or its index at print size. Check pages 12–16, 18, 24, and 29 at 100% zoom before export.
• Rational exponents. Set $\tfrac12$ and $\tfrac13$ as stacked fractions in exponents wherever the layout allows; $a^{1/2}$ with a slash is acceptable in tight tables but must not read as $a^1/2$. Pages 22, 25, 27, and 30 are the critical ones.
• Item order: a few items sit with their figures rather than in strict numeric order on a page, so every item on a page can be answered from that page's figure. The numbers still match the textbook exactly, and the answer key is in numerical order.
• Figure widths: fig1, fig2, fig3, fig4, fig5, fig6, fig7, fig8, fig9, and fig10 are full width on first appearance. fig2 and fig6 are wide two-panel images — set them full width and do not place text beside them. fig11-blank-radical-practice-tables.png takes a full page on its own. Review pages repeat fig1–fig10; repeat the image rather than asking students to flip back.
• The two load-bearing figures are fig8 and fig9. Figure 8 kills $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ with one counterexample; figure 9 forces $a^{1/2} = \sqrt{a}$ from the product law. Do not crop either conclusion box.
• Blank work space: explanation items 6, 11, 14, 20, 23, 31, 40, 46, 51, 60, 62, 74, 76, 77, 89, 97, 99, 100, and 104 need at least four ruled lines each.
• No division of radicals and no rationalizing appear anywhere in this chapter, by design. If a proof copy shows a rationalized denominator, it is a paste error and must come out.
• No variable radicands in any sum, difference, or product item. If a variable appears under a radical in a $\pm$ or $\times$ problem, it is a paste error.
• Only $\tfrac12$ and $\tfrac13$ appear as rational exponents. An exponent of $\tfrac14$, $\tfrac23$, or $\tfrac15$ is out of scope and must not appear.