8.6E: Exercises
Practice Makes Perfect
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. \(\dfrac{\sqrt[3]{162 x^{-3} y^{6}}}{\sqrt[3]{2 x^{3} y^{-2}}}\)
- Answer
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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. \(2 a b \sqrt[3]{2 a}\)
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. \(\dfrac{16}{\sqrt[4]{64 b^{2}}}\)
- Answer
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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. \(\dfrac{2 \sqrt[4]{3 a^{2}}}{a}\)
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. \(\dfrac{\sqrt{m}-\sqrt{3}}{\sqrt{m}+\sqrt{3}}\)
- Answer
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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. \(\dfrac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}\)
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- Simplify \(\sqrt{\dfrac{27}{3}}\) and explain all your steps.
- Simplify \(\sqrt{\dfrac{27}{5}}\) and explain all your steps.
- Why are the two methods of simplifying square roots different?
- Explain what is meant by the word rationalize in the phrase, "rationalize a denominator."
- Explain why multiplying \(\sqrt{2x}-3\) by its conjugate results in an epxression with no radicals.
- Explain why multiplying \(\dfrac{7}{\sqrt[3]{x}}\) by \(\dfrac{\sqrt[3]{x}}{\sqrt[3]{x}}\) does not rationalize the denominator.
- Answer
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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. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?