4.5E: Exercises

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Feb 26, 2021
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- 4.5: Add, Subtract, and Multiply Radical Expressions
- 4.6: Divide Radical Expressions

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Practice Makes Perfect

Exercise A: add and subtract radical expressions

In the following exercises, simplify. Assume all variables are greater than or equal to zero so that absolute values are not needed.

a. $8 \sqrt{2}-5 \sqrt{2}\quad$ b. $5 \sqrt[3]{m}+2 \sqrt[3]{m}\quad$ c. $8 \sqrt[4]{m}-2 \sqrt[4]{n}$

a. $7 \sqrt{2}-3 \sqrt{2}\quad$ b. $7 \sqrt[3]{p}+2 \sqrt[3]{p}\quad$ c. $5 \sqrt[3]{x}-3 \sqrt[3]{x}$

a. $3 \sqrt{5}+6 \sqrt{5}\quad$ b. $9 \sqrt[3]{a}+3 \sqrt[3]{a}\quad$ c. $5 \sqrt[4]{2 z}+\sqrt[4]{2 z}$

a. $4 \sqrt{5}+8 \sqrt{5} \quad$ b. $\sqrt[3]{m}-4 \sqrt[3]{m} \quad$ c. $\sqrt{n}+3 \sqrt{n}$

a. $3 \sqrt{2 a}-4 \sqrt{2 a}+5 \sqrt{2 a} \quad$ b. $5 \sqrt[4]{3 a b}-3 \sqrt[4]{3 a b}-2 \sqrt[4]{3 a b}$

a. $\sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b} \quad$ b. $8 \sqrt[4]{11 c d}+5 \sqrt[4]{11 c d}-9 \sqrt[4]{11 c d}$

a. $8 \sqrt{3 c}+2 \sqrt{3 c}-9 \sqrt{3 c} \quad$ b. $2 \sqrt[3]{4 p q}-5 \sqrt[3]{4 p q}+4 \sqrt[3]{4 p q}$

a. $3 \sqrt{5 d}+8 \sqrt{5 d}-11 \sqrt{5 d} \quad$ b. $11 \sqrt[3]{2 r s}-9 \sqrt[3]{2 r s}+3 \sqrt[3]{2 r s}$

a. $\sqrt{27}-\sqrt{75} \quad$ b. $\sqrt[3]{40}-\sqrt[3]{320} \quad$ c. $\frac{1}{2} \sqrt[4]{32}+\frac{2}{3} \sqrt[4]{162}$

a. $\sqrt{72}-\sqrt{98} \quad$ b. $\sqrt[3]{24}+\sqrt[3]{81} \quad$ c. $\frac{1}{2} \sqrt[4]{80}-\frac{2}{3} \sqrt[4]{405}$

a. $\sqrt{48}+\sqrt{27} \quad$ b. $\sqrt[3]{54}+\sqrt[3]{128} \quad$ c. $6 \sqrt[4]{5}-\frac{3}{2} \sqrt[4]{320}$

a. $\sqrt{45}+\sqrt{80} \quad$ b. $\sqrt[3]{81}-\sqrt[3]{192} \quad$ c. $\frac{5}{2} \sqrt[4]{80}+\frac{7}{3} \sqrt[4]{405}$

a. $\sqrt{72 a^{5}}-\sqrt{50 a^{5}} \quad$ b. $9 \sqrt[4]{80 p^{4}}-6 \sqrt[4]{405 p^{4}}$

a. $\sqrt{48 b^{5}}-\sqrt{75 b^{5}} \quad$ b. $8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}$

a. $\sqrt{80 c^{7}}-\sqrt{20 c^{7}} \quad$ b. $2 \sqrt[4]{162 r^{10}}+4 \sqrt[4]{32 r^{10}}$

a. $\sqrt{96 d^{9}}-\sqrt{24 d^{9}} \quad$ b. $5 \sqrt[4]{243 s^{6}}+2 \sqrt[4]{3 s^{6}}$

$3 \sqrt{128 y^{2}}+4 y \sqrt{162}-8 \sqrt{98 y^{2}}$

$3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}$

Answer

1. a. $3 \sqrt{2}$ b. $7 \sqrt[3]{m}$ c. $6 \sqrt[4]{m}$

3. a. $9 \sqrt{5}$ b. $12 \sqrt[3]{a}$ c. $6 \sqrt[4]{2 z}$

5. a. $4 \sqrt{2 a}$ b. $0$

7. a. $\sqrt{3c}$ b. $\sqrt[3]{4 p q}$

9. a. $-2 \sqrt{3}$ b. $-2 \sqrt[3]{5}$ c. $3 \sqrt[4]{2}$

11. a. $7 \sqrt{3}$ b. $7 \sqrt[3]{2}$ c. $3 \sqrt[4]{5}$

13. a. $a^{2} \sqrt{2 a}$ b. $0$

15. a. $2 c^{3} \sqrt{5 c}$ b. $14 r^{2} \sqrt[4]{2 r^{2}}$

17. $4 y \sqrt{2}$

Exercise B: multiply radical expressions

In the following exercises, simplify.

1. $(-2 \sqrt{3})(3 \sqrt{18})$
2. $(8 \sqrt[3]{4})(-4 \sqrt[3]{18})$

1. $(-4 \sqrt{5})(5 \sqrt{10})$
2. $(-2 \sqrt[3]{9})(7 \sqrt[3]{9})$

1. $(5 \sqrt{6})(-\sqrt{12})$
2. $(-2 \sqrt[4]{18})(-\sqrt[4]{9})$

1. $(-2 \sqrt{7})(-2 \sqrt{14})$
2. $(-3 \sqrt[4]{8})(-5 \sqrt[4]{6})$

1. $\left(4 \sqrt{12 z^{3}}\right)(3 \sqrt{9 z})$
2. $\left(5 \sqrt[3]{3 x^{3}}\right)\left(3 \sqrt[3]{18 x^{3}}\right)$

1. $\left(3 \sqrt{2 x^{3}}\right)\left(7 \sqrt{18 x^{2}}\right)$
2. $\left(-6 \sqrt[3]{20 a^{2}}\right)\left(-2 \sqrt[3]{16 a^{3}}\right)$

1. $\left(-2 \sqrt{7 z^{3}}\right)\left(3 \sqrt{14 z^{8}}\right)$
2. $\left(2 \sqrt[4]{8 y^{2}}\right)\left(-2 \sqrt[4]{12 y^{3}}\right)$

1. $\left(4 \sqrt{2 k^{5}}\right)\left(-3 \sqrt{32 k^{6}}\right)$
2. $\left(-\sqrt[4]{6 b^{3}}\right)\left(3 \sqrt[4]{8 b^{3}}\right)$

Answer

$-18 \sqrt{6}$

$-64 \sqrt[3]{9}$

$-30 \sqrt{2}$

$6 \sqrt[4]{2}$

$72 z^{2} \sqrt{3}$

$45 x^{2} \sqrt[3]{2}$

$-42 z^{5} \sqrt{2 z}$

$-8 y \sqrt[4]{6 y}$

Exercise C: use polynomial multiplication to multiply radical expressions

In the following exercises, multiply.

1. $\sqrt{7}(5+2 \sqrt{7})$
2. $\sqrt[3]{6}(4+\sqrt[3]{18})$

1. $\sqrt{11}(8+4 \sqrt{11})$
2. $\sqrt[3]{3}(\sqrt[3]{9}+\sqrt[3]{18})$

1. $\sqrt{11}(-3+4 \sqrt{11})$
2. $\sqrt[4]{3}(\sqrt[4]{54}+\sqrt[4]{18})$

1. $\sqrt{2}(-5+9 \sqrt{2})$
2. $\sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})$

$(7+\sqrt{3})(9-\sqrt{3})$

$(8-\sqrt{2})(3+\sqrt{2})$

1. $(9-3 \sqrt{2})(6+4 \sqrt{2})$
2. $(\sqrt[3]{x}-3)(\sqrt[3]{x}+1)$

1. $(3-2 \sqrt{7})(5-4 \sqrt{7})$
2. $(\sqrt[3]{x}-5)(\sqrt[3]{x}-3)$

1. $(1+3 \sqrt{10})(5-2 \sqrt{10})$
2. $(2 \sqrt[3]{x}+6)(\sqrt[3]{x}+1)$

1. $(7-2 \sqrt{5})(4+9 \sqrt{5})$
2. $(3 \sqrt[3]{x}+2)(\sqrt[3]{x}-2)$

$(\sqrt{3}+\sqrt{10})(\sqrt{3}+2 \sqrt{10})$

$(\sqrt{11}+\sqrt{5})(\sqrt{11}+6 \sqrt{5})$

$(2 \sqrt{7}-5 \sqrt{11})(4 \sqrt{7}+9 \sqrt{11})$

$(4 \sqrt{6}+7 \sqrt{13})(8 \sqrt{6}-3 \sqrt{13})$

1. $(3+\sqrt{5})^{2}$
2. $(2-5 \sqrt{3})^{2}$

1. $(4+\sqrt{11})^{2}$
2. $(3-2 \sqrt{5})^{2}$

1. $(9-\sqrt{6})^{2}$
2. $(10+3 \sqrt{7})^{2}$

1. $(5-\sqrt{10})^{2}$
2. $(8+3 \sqrt{2})^{2}$

$(4+\sqrt{2})(4-\sqrt{2})$

$(7+\sqrt{10})(7-\sqrt{10})$

$(4+9 \sqrt{3})(4-9 \sqrt{3})$

$(1+8 \sqrt{2})(1-8 \sqrt{2})$

$(12-5 \sqrt{5})(12+5 \sqrt{5})$

$(9-4 \sqrt{3})(9+4 \sqrt{3})$

$(\sqrt[3]{3 x}+2)(\sqrt[3]{3 x}-2)$

$(\sqrt[3]{4 x}+3)(\sqrt[3]{4 x}-3)$

Answer

$14+5 \sqrt{7}$

$4 \sqrt[3]{6}+3 \sqrt[3]{4}$

$44-3 \sqrt{11}$

$3 \sqrt[4]{2}+\sqrt[4]{54}$

5. $60+2 \sqrt{3}$

$30+18 \sqrt{2}$

$\sqrt[3]{x^{2}}-2 \sqrt[3]{x}-3$

$-54+13 \sqrt{10}$

$2 \sqrt[3]{x^{2}}+8 \sqrt[3]{x}+6$

11. $23+3 \sqrt{30}$

13. $-439-2 \sqrt{77}$

15.

$14+6 \sqrt{5}$

$79-20 \sqrt{3}$

17.

$87-18 \sqrt{6}$

$163+60 \sqrt{7}$

19. $14$

21. $-227$

23. $19$

25. $\sqrt[3]{9 x^{2}}-4$

Exercise D: mixed practice

$\frac{2}{3} \sqrt{27}+\frac{3}{4} \sqrt{48}$

$\sqrt{175 k^{4}}-\sqrt{63 k^{4}}$

$\frac{5}{6} \sqrt{162}+\frac{3}{16} \sqrt{128}$

$\sqrt[3]{24}+\sqrt[3]{ 81}$

$\frac{1}{2} \sqrt[4]{80}-\frac{2}{3} \sqrt[4]{405}$

$8 \sqrt[4]{13}-4 \sqrt[4]{13}-3 \sqrt[4]{13}$

$5 \sqrt{12 c^{4}}-3 \sqrt{27 c^{6}}$

$\sqrt{80 a^{5}}-\sqrt{45 a^{5}}$

$\frac{3}{5} \sqrt{75}-\frac{1}{4} \sqrt{48}$

$21 \sqrt[3]{9}-2 \sqrt[3]{9}$

$8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}$

$11 \sqrt{11}-10 \sqrt{11}$

$\sqrt{3} \cdot \sqrt{21}$

$(4 \sqrt{6})(-\sqrt{18})$

$(7 \sqrt[3]{4})(-3 \sqrt[3]{18})$

$\left(4 \sqrt{12 x^{5}}\right)\left(2 \sqrt{6 x^{3}}\right)$

$(\sqrt{29})^{2}$

$(-4 \sqrt{17})(-3 \sqrt{17})$

$(-4+\sqrt{17})(-3+\sqrt{17})$

$\left(3 \sqrt[4]{8 a^{2}}\right)\left(\sqrt[4]{12 a^{3}}\right)$

$(6-3 \sqrt{2})^{2}$

$\sqrt{3}(4-3 \sqrt{3})$

$\sqrt[3]{3}(2 \sqrt[3]{9}+\sqrt[3]{18})$

$(\sqrt{6}+\sqrt{3})(\sqrt{6}+6 \sqrt{3})$

Answer

1. $5\sqrt{3}$

3. $9\sqrt{2}$

5. $-\sqrt[4]{5}$

7. $10 c^{2} \sqrt{3}-9 c^{3} \sqrt{3}$

9. $2 \sqrt{3}$

11. $17 q^{2}$

13. $3 \sqrt{7}$

15. $-42 \sqrt[3]{9}$

17. $29$

19. $29-7 \sqrt{17}$

21. $72-36 \sqrt{2}$

23. $6+3 \sqrt[3]{2}$

Exercise E: writing exercises

Explain when a radical expression is in simplest form.
Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.
1. Explain why $(-\sqrt{n})^{2}$ is always non-negative, for $n \geq 0$ .
2. Explain why $-(\sqrt{n})^{2}$ is always non-positive, for $n \geq 0$ .
Use the binomial square pattern to simplify $(3+\sqrt{2})^{2}$ . Explain all your steps.

Answer

1. Answers will vary

3. Answers will vary

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is â€œI canâ€¦â€, the second is â€œConfidentlyâ€, the third is â€œWith some helpâ€, and the fourth is â€œNo, I donâ€™t get itâ€. Under the first column are the phrases â€œadd and subtract radical expressions.â€, â€œ multiply radical expressionsâ€, and â€œuse polynomial multiplication to multiply radical expressionsâ€. The other columns are left blank so that the learner may indicate their mastery level for each topic. — Figure 8.4.14

b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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Practice Makes Perfect

Exercise A: add and subtract radical expressions

Exercise B: multiply radical expressions

Exercise C: use polynomial multiplication to multiply radical expressions

Exercise D: mixed practice

Exercise E: writing exercises

Self Check

Practice Makes Perfect

Exercise A: add and subtract radical expressions

Exercise B: multiply radical expressions

Exercise C: use polynomial multiplication to multiply radical expressions

Exercise D: mixed practice

Exercise E: writing exercises

Self Check

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