Chapter 6
Section 6.1: Positive Integer Exponents
1. $9{{x}^{4}}$ 3. ${{2}^{15}}$ 5. $2{{x}^{3}}{{y}^{3}}$ 7. ${{2}^{50}}{{x}^{7}}{{y}^{17}}$ 9. ${{5}^{144}}{{x}^{48}}{{y}^{24}}$
11. ${{2}^{6}}{{x}^{14}}$ 13. ${{2}^{6}}{{x}^{7}}$ 15. ${{2}^{75}}{{x}^{7}}$ 17. ${{2}^{5}}\cdot {{3}^{8}}{{x}^{9}}{{y}^{5}}$ 19. ${{3}^{11}}{{x}^{13}}{{y}^{11}}$
21. ${{2}^{13x}}$ 23. $12{{x}^{5}}{{y}^{4}}$ 25. ${{2}^{4}}{{x}^{3}}{{y}^{10}}$ 27. ${{2}^{2}}x$ 29. $\tfrac{{{z}^{9}}}{{{x}^{2}}y}$
31. $\tfrac{-{{2}^{3}}}{{{x}^{3}}{{y}^{3}}}$ 33. ${{x}^{8m}}$ 35. ${{6}^{m}}$ 37. ${{x}^{2n}}$ 39. ${{2}^{3m}}{{x}^{5m}}$
41. ${{x}^{{{m}^{2}}-{{n}^{2}}}}$ 43. ${{x}^{a+2b}}$ 45. $0$ 47. ${{x}^{6{{a}^{2}}-a-2}}$ 49. ${{x}^{2a-2}}$ 51. ${{x}^{4a}}$
53. True 55. False 57. True 59. True 61. False
Section 6.2: Integer Exponents
1. $\tfrac{{{x}^{2}}}{2}$ 3. $\tfrac{18{{x}^{6}}}{y}$ 5. $\tfrac{{{x}^{4}}}{4{{y}^{6}}}$ 7. $\tfrac{{{2}^{4}}{{x}^{8}}}{{{y}^{12}}}$ 9. $\tfrac{{{y}^{6}}}{{{2}^{2}}{{x}^{4}}}$ 11. $\tfrac{{{x}^{2}}}{3}$ 13. ${{2}^{4}}\cdot {{3}^{2}}$
15. $\tfrac{2{{x}^{3}}}{9}$ 17. $\tfrac{{{x}^{4}}}{24{{y}^{2}}}$ 19. 4 21. ${{3}^{5}}{{K}^{6}}$ 23. $\tfrac{1}{{{x}^{13}}}$ 25. $\tfrac{{{y}^{2}}}{{{x}^{4}}}$ 27. $\tfrac{{{x}^{8}}}{{{y}^{8}}}$
29. ${{x}^{m}}$ 31. $\tfrac{a}{1+ab}$ 33. $x+1$ 35. $\tfrac{1}{{{m}^{6}}}$ 37. $\tfrac{1}{{{(2a-b)}^{2}}}$ 39. $\tfrac{1}{x}$
41. $a{{b}^{2}}$ 43. ${{2}^{-1}}{{x}^{2}}{{y}^{-2}}$ 45. ${{2}^{3}}\cdot 3{{x}^{-4}}{{y}^{6}}$ 47. ${{2}^{-1}}\cdot {{3}^{2}}{{x}^{-2}}{{y}^{-12}}$
49. a. True b. False c. False d. True e. False f. False
g. True h. True i. True j. False k. False l. False
51. a. ${{2}^{3}}{{x}^{6}}{{y}^{9}}$ b. ${{2}^{2}}{{x}^{2}}$ c. ${{x}^{-2}}{{y}^{2}}$ d. $2{{x}^{2}}y$ e. ${{x}^{6}}{{y}^{-2}}$
f. $x{{y}^{5}}$ g. ${{2}^{5}}{{x}^{4}}{{y}^{2}}$ h. ${{2}^{6}}{{x}^{2}}$
Section 6.3: Radical Simplification 1
1. $5\sqrt{2}$ 3. $10\sqrt{2}$ 5. $30\sqrt{10}$ 7. ${{3}^{4}}\cdot {{5}^{2}}\cdot 7\sqrt{2}$ 9. $720\sqrt{7}$
11. $\tfrac{2\sqrt{2}}{5}$ 13. $\tfrac{\sqrt{6}}{3}$ 15. $\tfrac{\sqrt{15}}{5}$ 17. $2xy\sqrt{2y}$ 19. ${{x}^{2}}{{y}^{2}}\sqrt{5x}$
21. $2a{{b}^{2}}{{c}^{3}}\sqrt{5}$ 23. ${{2}^{4}}{{x}^{2}}{{y}^{2}}$ 25. $2\cdot {{3}^{2}}{{a}^{3}}{{b}^{5}}\sqrt{b}$ 27. $4{{x}^{2}}\sqrt{y}$ 29. $\tfrac{\sqrt{a}}{3b}$
31. $\sqrt{a}$ 33. $\tfrac{\sqrt{2a}}{2a}$ 35. $\tfrac{\sqrt{5a}}{{{a}^{2}}}$ 37. $2abc\sqrt{2c}$ 39. $4\pm \sqrt{19}$
41. $\tfrac{2\pm \sqrt{7}}{3}$ 43. $6\pm \sqrt{35}$ 45. $36x{{y}^{4}}z$ 47. ${{x}^{10}}\sqrt{x}$ 49. ${{2}^{24}}\cdot {{3}^{18}}\sqrt{6}$
51. $2{{x}^{7}}\sqrt{30}$
Section 6.4: Radical Simplification 2
1. $3\sqrt{2}$ 3. $6\sqrt{3}-6\sqrt{2}$ 5. $\tfrac{\sqrt{2}}{2}-\tfrac{\sqrt{3}}{3}$ 7. $6\sqrt{2x}$ 9. $\tfrac{2\sqrt{15}}{3}$
11. $\tfrac{6\sqrt{6}}{6}$ 13. $\tfrac{5\sqrt{2}}{12}$ 15. $\tfrac{11\sqrt{2}}{6}$ 17. 47 19. 2
21. 58 23. $18-8\sqrt{2}$ 25. $5-2\sqrt{6}$ 27. $11\sqrt{3}-4\sqrt{2}$ 29. $\tfrac{1+\sqrt{5}}{2}$
31. $\sqrt{5}$ 33. $\sqrt{5}-\sqrt{x}$ 35. $\sqrt{a}-\sqrt{b}$ 37. $\tfrac{11+6\sqrt{2}}{7}$ 39. $5-2\sqrt{6}$
41. $\tfrac{3+\sqrt{5}}{2}$ 43. $2\sqrt{2}$ 45. $2\sqrt{5}$ 47. 20
Section 6.5: Rational Exponents
1. 4 3. 5 5. 16 7. 16 9. 625 11. ${\scriptstyle{}^{{{x}^{4}}}\!\!\diagup\!\!{}_{{{y}^{18}}}\;}$
13. ${\scriptstyle{}^{1}\!\!\diagup\!\!{}_{8}\;}$ 15. ${\scriptstyle{}^{8}\!\!\diagup\!\!{}_{27}\;}$ 17. $4{{a}^{2}}{{n}^{{\scriptstyle{}^{1}\!\!\diagup\!\!{}_{3}\;}}}{{p}^{-{\scriptstyle{}^{1}\!\!\diagup\!\!{}_{2}\;}}}$ 19. ${\scriptstyle{}^{9{{y}^{2}}}\!\!\diagup\!\!{}_{{{x}^{4}}}\;}$ 21. ${{x}^{4}}$
23. ${\scriptstyle{}^{1}\!\!\diagup\!\!{}_{4}\;}$ 25. ${\scriptstyle{}^{x}\!\!\diagup\!\!{}_{{{y}^{2}}}\;}$ 27. 9 29. ${\scriptstyle{}^{3a}\!\!\diagup\!\!{}_{b}\;}$ 31. $mn$ 33. $x-y$
35. $\tfrac{a+2\sqrt{ab}+b}{ab}$ 37. ${{2}^{{\scriptstyle{}^{17}\!\!\diagup\!\!{}_{6}\;}}}{{t}^{{\scriptstyle{}^{13}\!\!\diagup\!\!{}_{6}\;}}}$ 39. ${{(3K)}^{{\scriptstyle{}^{11}\!\!\diagup\!\!{}_{12}\;}}}$ 41. ${{x}^{{\scriptstyle{}^{1}\!\!\diagup\!\!{}_{6}\;}}}$ 43. $4t$
45. ${{(2t)}^{{\scriptstyle{}^{29}\!\!\diagup\!\!{}_{12}\;}}}$ 47. ${{(3t)}^{{\scriptstyle{}^{5}\!\!\diagup\!\!{}_{4}\;}}}$ 49. True 51. True 53. False
Section 6.6: Radical Equations
1. 6 3. 6 5. 4 7. 0, 1 9. 3 11. 3
13. 4 15. 5 17. $-1$ 19. 2, 5 21. $-2$
Section 6.7: Complex Fractions
1. $\tfrac{x}{1-2x}$ 3. $\tfrac{ab}{b-2a}$ 5. $\tfrac{2x-3}{3x-1}$ 7. $\tfrac{6}{5}$ 9. $\tfrac{3m-1}{{{m}^{2}}-9}$ 11. $-(x+y)$
13. $\tfrac{1}{2x}$ 15. $\tfrac{x-1}{x}$ 17. $\tfrac{2x+y}{x}$ 19. $\tfrac{-x+2}{2x-1}$ 21. $\tfrac{1}{x}$
Section 6.8: Miscellaneous Exponent and Radical Problems
1. ${\scriptstyle{}^{{{a}^{8}}}\!\!\diagup\!\!{}_{{{b}^{12}}}\;}$ 3. ${\scriptstyle{}^{{{3}^{4}}{{b}^{12}}}\!\!\diagup\!\!{}_{{{a}^{8}}}\;}$ 5. ${\scriptstyle{}^{{{x}^{5}}}\!\!\diagup\!\!{}_{{{y}^{4}}}\;}$ 7. $81{{x}^{12}}{{y}^{6}}$ 9. ${\scriptstyle{}^{{{b}^{12}}}\!\!\diagup\!\!{}_{8{{a}^{3}}}\;}$
11. ${\scriptstyle{}^{{{a}^{2}}{{b}^{2}}}\!\!\diagup\!\!{}_{\left( {{b}^{2}}-{{a}^{2}} \right)}\;}$ 13. ${\scriptstyle{}^{9}\!\!\diagup\!\!{}_{8{{y}^{2}}}\;}$ 15. ${\scriptstyle{}^{{{y}^{10}}}\!\!\diagup\!\!{}_{{{3}^{3}}{{x}^{4}}}\;}$ 17. $2x+2\sqrt{{{x}^{2}}-{{y}^{2}}}$
19. $26\sqrt{2}$ 21. $15\sqrt{6x}$ 23. $4\sqrt{3}$ 25. $2(4-\sqrt{13})$ 27. $2(\sqrt{7}+\sqrt{3})$
29. $2a{{b}^{2}}{{c}^{4}}\cdot \sqrt[4]{4{{a}^{2}}}$ 31. ${\scriptstyle{}^{4{{c}^{5}}}\!\!\diagup\!\!{}_{ab}\;}$ 33. $6\sqrt{5}$ 35. ${{3}^{301}}$
37. a. $16{{x}^{12}}$ b. ${\scriptstyle{}^{1}\!\!\diagup\!\!{}_{{{x}^{8}}}\;}$ c. ${\scriptstyle{}^{1}\!\!\diagup\!\!{}_{\left( {{2}^{8}}{{x}^{8}} \right)}\;}$ d. ${{2}^{4}}{{x}^{4}}$
39. a. $-1$ b. 2 c. 7