Section 6.8
Miscellaneous Exponent and Radical Problems
Simplify each of the following.
\begin{array}{l l l}
1. \;\;\;{{\left( {{a}^{-2}}{{b}^{3}} \right)}^{-4}}\;\;\; & 2. \;\;\;{{\left( 2{{x}^{2}}{{y}^{-3}} \right)}^{4}}\;\;\; & 3. \;\;\;{{\left( {{3}^{-1}}{{a}^{2}}{{b}^{-3}} \right)}^{-4}}\;\;\; \\
4. \;\;\;{{\left( {{3}^{-1}}a{{b}^{-2}} \right)}^{2}}\cdot {{\left( 9{{a}^{-1}}{{b}^{2}} \right)}^{-3}}\;\;\; & 5. \;\;\;{{\left( 2{{x}^{2}}{{y}^{-1}} \right)}^{2}}\cdot {{\left( 4{{x}^{-1}}{{y}^{2}} \right)}^{-1}}\;\;\; & 6. \;\;\;{{\left( {{4}^{2}}{{a}^{-1}}{{b}^{3}} \right)}^{2}}\cdot {{\left( 2a{{b}^{-2}} \right)}^{3}}\;\;\; \\
7. \;\;\;{{\left( \frac{3{{x}^{-4}}{{y}^{-2}}}{27{{x}^{2}}y} \right)}^{-2}}\;\;\; & 8. \;\;\;{{\left( \frac{{{4}^{2}}{{a}^{-1}}{{b}^{3}}}{2a{{b}^{-2}}} \right)}^{2}}\;\;\; & 9. \;\;\;{{\left( \frac{8{{a}^{3}}{{b}^{-2}}}{{{2}^{2}}{{a}^{2}}{{b}^{2}}} \right)}^{-3}}\;\;\; \\
10. \;\;\;{{\left( {{3}^{-1}}-{{x}^{-1}} \right)}^{-1}}\;\;\; & 11. \;\;\;{{\left( {{a}^{-2}}-{{b}^{-2}} \right)}^{-1}}\;\;\; & 12. \;\;\;{{\left( {{a}^{-1/m}}-{{b}^{-1/m}} \right)}^{-m}}\;\;\; \\
13. \;\;\;{{\left( \frac{2x{{y}^{0}}}{3{{x}^{-1}}{{y}^{-2}}} \right)}^{-2}}\cdot \left( \frac{5x{{y}^{-2}}}{10{{x}^{-3}}} \right)\;\;\; & 14. \;\;\;{{\left( \frac{2xy}{3{{y}^{-2}}} \right)}^{2}}\cdot {{\left( \frac{3{{y}^{-2}}}{6} \right)}^{-2}}\;\;\; & 15. \;\;\;{{\left( \frac{2xy}{3{{y}^{-2}}} \right)}^{2}}\cdot {{\left( \frac{10{{x}^{0}}{{y}^{-2}}}{5{{x}^{-3}}} \right)}^{-2}}\;\;\; \\
16. \;\;\;\frac{{{\left( {{a}^{1/m}}\cdot {{b}^{1/n}} \right)}^{mn}}}{{{\left( {{a}^{-m/2}}{{b}^{n/2}} \right)}^{2}}}\;\;\; & 17. \;\;\;{{\left( \sqrt{x+y}+\sqrt{x-y} \right)}^{2}}\;\;\; & 18. \;\;\;{{\left( {{a}^{1/2}}+{{b}^{1/2}} \right)}^{2}}-{{\left( {{a}^{1/2}}-{{b}^{1/2}} \right)}^{2}}\;\;\; \\
19. \;\;\;\sqrt{72}-\sqrt{288}+\sqrt{2048}\;\;\; & 20. \;\;\;3\sqrt{20}+2\sqrt{45}-\sqrt{80}\;\;\; & 21. \;\;\;\tfrac{1}{x}\sqrt{54{{x}^{3}}}+2\sqrt{294x}-x\sqrt{\tfrac{24}{x}}\;\;\; \\
22. \;\;\;\tfrac{8}{\sqrt{6}}\;\;\; & 23. \;\;\;\tfrac{12}{\sqrt{3}}\;\;\; & 24. \;\;\;\sqrt{\tfrac{8}{5x}}\;\;\; \\
25. \;\;\;\tfrac{6}{4+\sqrt{13}}\;\;\; & 26. \;\;\;\tfrac{8}{\sqrt{5}+\sqrt{3}}\;\;\; & 27. \;\;\;\tfrac{8}{\sqrt{7}-\sqrt{3}}\;\;\; \\
28. \;\;\;\sqrt[3]{72{{a}^{6}}{{b}^{9}}{{c}^{13}}}\;\;\; & 29. \;\;\;\sqrt[4]{64{{a}^{6}}{{b}^{8}}{{c}^{16}}}\;\;\; & 30. \;\;\;\sqrt{12{{x}^{3}}}-\sqrt{27{{x}^{5}}}+\sqrt{\tfrac{108}{x}}\;\;\; \\
31. \;\;\;\sqrt[3]{\frac{64{{a}^{-2}}{{b}^{-1}}{{c}^{16}}}{a{{b}^{2}}c}}\;\;\; & 32. \;\;\;\sqrt{{{10}^{2}}-{{6}^{2}}}\;\;\; & 33. \;\;\;\sqrt{{{12}^{2}}+{{6}^{2}}}\;\;\; \\
34. \;\;\;{{2}^{200}}+{{2}^{200}}\;\;\; & 35. \;\;\;{{3}^{300}}+{{3}^{300}}+{{3}^{300}}\;\;\; & 36. \;\;\;{{5}^{500}}+{{5}^{500}}+{{5}^{500}}+{{5}^{500}}+{{5}^{500}}\;\;\;
\end{array}
37. Simplify each of the following. (The answers are all different.)
a. $2{{x}^{0}}\cdot 2{{x}^{2}}\cdot 2{{x}^{4}}\cdot 2{{x}^{6}}$ b. $\left( 2{{x}^{0}}\cdot 2{{x}^{2}} \right)\div \left( 2{{x}^{4}}\cdot 2{{x}^{6}} \right)$
c. $\left( {{\left( 2x \right)}^{0}}\cdot {{\left( 2x \right)}^{2}} \right)\div \left( {{\left( 2x \right)}^{4}}\cdot {{\left( 2x \right)}^{6}} \right)$ d. ${{\left( 2x \right)}^{0}}\cdot {{\left( 2x \right)}^{2}}\div {{\left( 2x \right)}^{4}}\cdot {{\left( 2x \right)}^{6}}$
38. Perform the indicated operations and express the result in scientific notation.
a. $\left( 1.7\times {{10}^{14}} \right)\cdot \left( 8.2\times {{10}^{27}} \right)$ b. $\frac{1.2\times {{10}^{-34}}}{1.6\times {{10}^{17}}}$
39. Solve each of the following equations for x.
a. $5+\sqrt{x+2}=6 \qquad$ b. $4-\sqrt{x+2}=x \qquad$ c. $4+\sqrt{x+2}=x$