10.6E: Exercises
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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. \(\dfrac{\sqrt[3]{162 x^{-3} y^{6}}}{\sqrt[3]{2 x^{3} y^{-2}}}\)
- 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. \(2 a b \sqrt[3]{2 a}\)
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. \(\dfrac{16}{\sqrt[4]{64 b^{2}}}\)
- 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. \(\dfrac{2 \sqrt[4]{3 a^{2}}}{a}\)
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. \(\dfrac{\sqrt{m}-\sqrt{3}}{\sqrt{m}+\sqrt{3}}\)
- 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. \(\dfrac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}\)
Exercise SET D: writing exercises
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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?