# 8.4E: Exercises


### Practice Makes Perfect

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

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

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

3.
4.

1. $$(-4 \sqrt{5})(5 \sqrt{10})$$

2. $$(-2 \sqrt[3]{9})(7 \sqrt[3]{9})$$

3.
4.

1. $$(5 \sqrt{6})(-\sqrt{12})$$

2. $$(-2 \sqrt[4]{18})(-\sqrt[4]{9})$$

3.
4.

1. $$(-2 \sqrt{7})(-2 \sqrt{14})$$

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

3.
4.

1. $$\left(4 \sqrt{12 z^{3}}\right)(3 \sqrt{9 z})$$

2. $$\left(5 \sqrt[3]{3 x^{3}}\right)\left(3 \sqrt[3]{18 x^{3}}\right)$$

3.
4.

1. $$\left(3 \sqrt{2 x^{3}}\right)\left(7 \sqrt{18 x^{2}}\right)$$

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

3.
4.

1. $$\left(-2 \sqrt{7 z^{3}}\right)\left(3 \sqrt{14 z^{8}}\right)$$

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

3.
4.

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

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

3.
4.

1.

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

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

3.
4.

3.

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

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

3.
4.

5.

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

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

3.
4.

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. $$\sqrt[3]{6}(4+\sqrt[3]{18})$$

3.
4.

1. $$\sqrt{11}(8+4 \sqrt{11})$$

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

3.
4.

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

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

3.
4.

1. $$\sqrt{2}(-5+9 \sqrt{2})$$

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

3.
4.

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

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

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

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

3.
4.

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

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

3.
4.

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

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

3.
4.

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

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

3.
4.

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

3.
4.

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

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

3.
4.

1. $$(9-\sqrt{6})^{2}$$

2. $$(10+3 \sqrt{7})^{2}$$

3.
4.

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

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

3.
4.

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

1.

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

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

3.
4.

3.

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

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

3.
4.

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

7.

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

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

3.
4.

9.

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

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

3.
4.

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

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.