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# 8.4E: Exercises


### Practice Makes Perfect

Exercise A: add and subtract radical expressions

In the following exercises, simplify.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

17. $$3 \sqrt{128 y^{2}}+4 y \sqrt{162}-8 \sqrt{98 y^{2}}$$

18. $$3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}$$
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. $$-2 \sqrt{3}$$ 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

1.

1. $$-18 \sqrt{6}$$

2. $$-64 \sqrt[3]{9}$$

3.

1. $$-30 \sqrt{2}$$

2. $$6 \sqrt[4]{2}$$

5.

1. $$72 z^{2} \sqrt{3}$$

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

7.

1. $$-42 z^{5} \sqrt{2 z}$$

2. $$-8 y \sqrt[4]{6 y}$$

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. $$\sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})$$

1. $$(7+\sqrt{3})(9-\sqrt{3})$$

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

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

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

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

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

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

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. $$(8+3 \sqrt{2})^{2}$$

7. $$(4+\sqrt{2})(4-\sqrt{2})$$

8. $$(7+\sqrt{10})(7-\sqrt{10})$$

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

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

11. $$(12-5 \sqrt{5})(12+5 \sqrt{5})$$

12. $$(9-4 \sqrt{3})(9+4 \sqrt{3})$$

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

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

1.

1. $$14+5 \sqrt{7}$$

2. $$4 \sqrt[3]{6}+3 \sqrt[3]{4}$$

3.

1. $$44-3 \sqrt{11}$$

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

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

7.

1. $$30+18 \sqrt{2}$$

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

9.

1. $$-54+13 \sqrt{10}$$

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

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

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

15.

1. $$14+6 \sqrt{5}$$

2. $$79-20 \sqrt{3}$$

17.

1. $$87-18 \sqrt{6}$$

2. $$163+60 \sqrt{7}$$

19. $$14$$

21. $$-227$$

23. $$19$$

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

Exercise D: mixed practice

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

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

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

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

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

6. $$8 \sqrt[4]{13}-4 \sqrt[4]{13}-3 \sqrt[4]{13}$$

7. $$5 \sqrt{12 c^{4}}-3 \sqrt{27 c^{6}}$$

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

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

10. $$21 \sqrt[3]{9}-2 \sqrt[3]{9}$$

11. $$8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}$$

12. $$11 \sqrt{11}-10 \sqrt{11}$$

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

14. $$(4 \sqrt{6})(-\sqrt{18})$$

15. $$(7 \sqrt[3]{4})(-3 \sqrt[3]{18})$$

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

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

18. $$(-4 \sqrt{17})(-3 \sqrt{17})$$

19. $$(-4+\sqrt{17})(-3+\sqrt{17})$$

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

21. $$(6-3 \sqrt{2})^{2}$$

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

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

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

1. Explain when a radical expression is in simplest form.
2. 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 $$n \geq 0$$.
3. 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.

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?