$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 8.6E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

### Practice Makes Perfect

##### Exercise SET A: divide square roots

In the following exercises, simplify.

1. a. $$\dfrac{\sqrt{128}}{\sqrt{72}}\quad$$ b. $$\dfrac{\sqrt[3]{128}}{\sqrt[3]{54}}$$

2. a. $$\dfrac{\sqrt{48}}{\sqrt{75}}\quad$$ b. $$\dfrac{\sqrt[3]{81}}{\sqrt[3]{24}}$$

3. a.$$\dfrac{\sqrt{200 m^{5}}}{\sqrt{98 m}}\quad$$ b. $$\dfrac{\sqrt[3]{54 y^{2}}}{\sqrt[3]{2 y^{5}}}$$

4. a. $$\dfrac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}\quad$$ b. $$\dfrac{\sqrt[3]{54 y}}{\sqrt[3]{16 y^{4}}}$$

5. a. $$\dfrac{\sqrt{75 r^{3}}}{\sqrt{108 r^{7}}}\quad$$ b. $$\dfrac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{81 x^{4}}}$$

6. a. $$\dfrac{\sqrt{196 q}}{\sqrt{484 q^{5}}}\quad$$ b. $$\dfrac{\sqrt[3]{16 m^{4}}}{\sqrt[3]{54 m}}$$

7. a. $$\dfrac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}}\quad$$ b. $$\dfrac{\sqrt[3]{-16 a^{4} b^{-2}}}{\sqrt[3]{2 a^{-2} b}}$$

8. a. $$\dfrac{\sqrt{98 r s^{10}}}{\sqrt{2 r^{3} s^{4}}}\quad$$ b. $$\dfrac{\sqrt[3]{-375 y^{4} z^{2}}}{\sqrt[3]{3 y^{-2} z^{4}}}$$

9. a. $$\dfrac{\sqrt{320 m n^{-5}}}{\sqrt{45 m^{-7} n^{3}}}\quad$$ b. $$\dfrac{\sqrt[3]{16 x^{4} y^{-2}}}{\sqrt[3]{-54 x^{-2} y^{4}}}$$

10. a. $$\dfrac{\sqrt{810 c^{-3} d^{7}}}{\sqrt{1000 c d}}\quad$$ b. $$\dfrac{\sqrt[3]{24 a^{7} b^{-1}}}{\sqrt[3]{-81 a^{-2} b^{2}}}$$

11. $$\dfrac{\sqrt{56 x^{5} y^{4}}}{\sqrt{2 x y^{3}}}$$

12. $$\dfrac{\sqrt{72 a^{3} b^{6}}}{\sqrt{3 a b^{3}}}$$

13. $$\dfrac{\sqrt[3]{48 a^{3} b^{6}}}{\sqrt[3]{3 a^{-1} b^{3}}}$$

14. $$\dfrac{\sqrt[3]{162 x^{-3} y^{6}}}{\sqrt[3]{2 x^{3} y^{-2}}}$$

1. a. $$\dfrac{4}{3}$$ b. $$\dfrac{4}{3}$$

3. a. $$\dfrac{10 m^{2}}{7}$$ b. $$\dfrac{3}{y}$$

5. a. $$\dfrac{5}{6 r^{2}}$$ b. $$\dfrac{2x}{3}$$

7. a. $$\dfrac{6 p}{q^{2}}$$ b. $$-\dfrac{2 a^{2}}{b}$$

9. a. $$\dfrac{8 m^{4}}{3 n^{4}}$$ b. $$-\dfrac{2 x^{2}}{3 y^{2}}$$

11. $$4 x^{4} \sqrt{7 y}$$

13. $$2 a b \sqrt[3]{2 a}$$

##### Exercise SET B: Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

15. a. $$\dfrac{10}{\sqrt{6}}\quad$$ b. $$\sqrt{\dfrac{4}{27}}\quad$$ c. $$\dfrac{10}{\sqrt{5 x}}$$

16. a. $$\dfrac{8}{\sqrt{3}}\quad$$ b. $$\sqrt{\dfrac{7}{40}}\quad$$ c. $$\dfrac{8}{\sqrt{2 y}}$$

17. a. $$\dfrac{6}{\sqrt{7}}\quad$$ b. $$\sqrt{\dfrac{8}{45}}\quad$$ c. $$\dfrac{12}{\sqrt{3 p}}$$

18. a. $$\dfrac{4}{\sqrt{5}}\quad$$ b. $$\sqrt{\dfrac{27}{80}}\quad$$ c. $$\dfrac{18}{\sqrt{6 q}}$$

19. a. $$\dfrac{1}{\sqrt[3]{5}}\quad$$ b. $$\sqrt[3]{\dfrac{5}{24}}\quad$$ c. $$\dfrac{4}{\sqrt[3]{36 a}}$$

20. a. $$\dfrac{1}{\sqrt[3]{3}}\quad$$ b. $$\sqrt[3]{\dfrac{5}{32}}\quad$$ c. $$\dfrac{7}{\sqrt[3]{49 b}}$$

21. a. $$\dfrac{1}{\sqrt[3]{11}}\quad$$ b. $$\sqrt[3]{\dfrac{7}{54}}\quad$$ c. $$\dfrac{3}{\sqrt[3]{3 x^{2}}}$$

22. a. $$\dfrac{1}{\sqrt[3]{13}}\quad$$ b. $$\sqrt[3]{\dfrac{3}{128}}\quad$$ c. $$\dfrac{3}{\sqrt[3]{6 y^{2}}}$$

23. a. $$\dfrac{1}{\sqrt[4]{7}}\quad$$ b. $$\sqrt[4]{\dfrac{5}{32}}\quad$$ c. $$\dfrac{4}{\sqrt[4]{4 x^{2}}}$$

24. a. $$\dfrac{1}{\sqrt[4]{4}}\quad$$ b. $$\sqrt[4]{\dfrac{9}{32}}\quad$$ c. $$\dfrac{6}{\sqrt[4]{9 x^{3}}}$$

25. a. $$\dfrac{1}{\sqrt[4]{9}}\quad$$ b. $$\sqrt[4]{\dfrac{25}{128}}\quad$$ c. $$\dfrac{6}{\sqrt[4]{27 a}}$$

26. a. $$\dfrac{1}{\sqrt[4]{8}}\quad$$ b. $$\sqrt[4]{\dfrac{27}{128}}\quad$$ c. $$\dfrac{16}{\sqrt[4]{64 b^{2}}}$$

15. a. $$\dfrac{5 \sqrt{6}}{3}$$ b. $$\dfrac{2 \sqrt{3}}{9}$$ c. $$\dfrac{2 \sqrt{5 x}}{x}$$

17. a. $$\dfrac{6 \sqrt{7}}{7}$$ b. $$\dfrac{2 \sqrt{10}}{15}$$ c. $$\dfrac{4 \sqrt{3 p}}{p}$$

19. a. $$\dfrac{\sqrt[3]{25}}{5}$$ b. $$\dfrac{\sqrt[3]{45}}{6}$$ c. $$\dfrac{2 \sqrt[3]{6 a^{2}}}{3 a}$$

21. a. $$\dfrac{\sqrt[3]{121}}{11}$$ b. $$\dfrac{\sqrt[3]{28}}{6}$$ c. $$\dfrac{\sqrt[3]{9 x}}{x}$$

23. a. $$\dfrac{\sqrt[4]{343}}{7}$$ b. $$\dfrac{\sqrt[4]{40}}{4}$$ c. $$\dfrac{2 \sqrt[4]{4 x^{2}}}{x}$$

25. a. $$\dfrac{\sqrt[4]{9}}{3}$$ b. $$\dfrac{\sqrt[4]{50}}{4}$$ c. $$\dfrac{2 \sqrt[4]{3 a^{2}}}{a}$$

##### Exercise SET C: Rationalize a Two Term Denominator

In the following exercises, simplify.

27. $$\dfrac{8}{1-\sqrt{5}}$$

28. $$\dfrac{7}{2-\sqrt{6}}$$

29. $$\dfrac{6}{3-\sqrt{7}}$$

30. $$\dfrac{5}{4-\sqrt{11}}$$

31. $$\dfrac{\sqrt{3}}{\sqrt{m}-\sqrt{5}}$$

32. $$\dfrac{\sqrt{5}}{\sqrt{n}-\sqrt{7}}$$

33. $$\dfrac{\sqrt{2}}{\sqrt{x}-\sqrt{6}}$$

34. $$\dfrac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}$$

35. $$\dfrac{\sqrt{r}+\sqrt{5}}{\sqrt{r}-\sqrt{5}}$$

36. $$\dfrac{\sqrt{s}-\sqrt{6}}{\sqrt{s}+\sqrt{6}}$$

37. $$\dfrac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}$$

38. $$\dfrac{\sqrt{m}-\sqrt{3}}{\sqrt{m}+\sqrt{3}}$$

27. $$-2(1+\sqrt{5})$$

29. $$3(3+\sqrt{7})$$

31. $$\dfrac{\sqrt{3}(\sqrt{m}+\sqrt{5})}{m-5}$$

33. $$\dfrac{\sqrt{2}(\sqrt{x}+\sqrt{6})}{x-6}$$

35. $$\dfrac{(\sqrt{r}+\sqrt{5})^{2}}{r-5}$$

37. $$\dfrac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}$$

##### Exercise SET D: writing exercises
1. Simplify $$\sqrt{\dfrac{27}{3}}$$ and explain all your steps.
2. Simplify $$\sqrt{\dfrac{27}{5}}$$ and explain all your steps.
3. Why are the two methods of simplifying square roots different?
1. Explain what is meant by the word rationalize in the phrase, "rationalize a denominator."
2. Explain why multiplying $$\sqrt{2x}-3$$ by its conjugate results in an epxression with no radicals.
3. Explain why multiplying $$\dfrac{7}{\sqrt[3]{x}}$$ by $$\dfrac{\sqrt[3]{x}}{\sqrt[3]{x}}$$ does not rationalize the denominator.