Section 6.5
Rational Exponents
We extend the realm of exponents one more time, to allow rational numbers as exponents. We want to preserve our existing rules for exponents. Since ${{a}^{1/n}}=\sqrt[n]{a}$ and ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ for the exponents we have dealt with so far, we should have ${{\left( {{a}^{1/n}} \right)}^{n}}={{a}^{\left( \tfrac{1}{n} \right)\cdot n}}={{a}^{1}}=a$. Thus, if ${{a}^{1/n}}$ is to have any meaning, it must be a number that can be raised to the ${{n}^{\text{th}}}$ power to produce then number $a$; that is, ${{a}^{1/n}}$ is the ${{n}^{\text{th}}}$ root of $a$. We therefore have the following definition.
Definition: ${{a}^{1/n}}=\sqrt[n]{a}$.
- If $a$ is a positive number, then ${{a}^{1/n}}$ is the positive ${{n}^{\text{th}}}$ root of $a$.
- If $a$ is a negative number and $n$ is even, then ${{a}^{1/n}}$ is undefined.
- If $a$ is a negative number and $n$ is odd, then ${{a}^{1/n}}$ will be a negative number.
\begin{align}
& \text{a. }{{4}^{1/2}}=\sqrt{4}=2 \\ \\
& \text{b. }{{8}^{2/3}}={{\left( {{8}^{1/3}} \right)}^{2}}={{\left( \sqrt[3]{8} \right)}^{2}}={{2}^{2}}=4 \\ \\
& \text{c. }\sqrt{2}\cdot \sqrt[3]{2}={{2}^{1/2}}\cdot {{2}^{1/3}}={{2}^{5/6}} \\ \\
& \text{d. }{{\left( \sqrt[3]{{{x}^{2}}}\cdot \sqrt[4]{y} \right)}^{12}}={{\left( {{x}^{2/3}}{{y}^{1/4}} \right)}^{12}}={{x}^{8}}{{y}^{3}} \\ \\
& \text{e. }{{\left( {{x}^{1/2}}+{{y}^{1/2}} \right)}^{2}}=x+2{{x}^{1/2}}{{y}^{1/2}}+y \\ \\
& \text{f. }{{100}^{3/2}}={{\left( {{100}^{1/2}} \right)}^{3}}={{10}^{3}}=1000 \\ \\
& \text{g. }{{4}^{-3/2}}={{\left( {{4}^{3/2}} \right)}^{-1}}={{8}^{-1}}=\frac{1}{8} \\ \\
& \text{h. }{{2}^{3/2}}={{2}^{1}}\cdot {{2}^{1/2}}=2\sqrt{2} \\ \\
& \text{i. }\sqrt[3]{{{a}^{2}}b}={{\left( {{a}^{2}}b \right)}^{1/3}}={{a}^{2/3}}{{b}^{1/3}} \\
\end{align}
Basic fractional exponents:
Negative fractional exponent examples 2:
Fractional exponents with numerators other than 1:
Exercises.
Write each of the following in simplest form.
\begin{array}{l l l}
1. \;\;\;{{16}^{1/2}}\;\;\; & 2. \;\;\;{{64}^{1/3}}\;\;\; & 3. \;\;\;{{125}^{1/3}}\;\;\; \\
4. \;\;\;{{8}^{1/3}}\;\;\; & 5. \;\;\;{{64}^{2/3}}\;\;\; & 6. \;\;\;{{4}^{3/2}}\;\;\; \\
7. \;\;\;{{8}^{4/3}}\;\;\; & 8. \;\;\;{{9}^{5/2}}\;\;\; & 9. \;\;\;{{125}^{4/3}}\;\;\; \\
10. \;\;\;{{\left( {{x}^{1/3}}\cdot {{y}^{1/2}} \right)}^{6}}\;\;\; & 11. \;\;\;{{\left( {{x}^{1/3}}\cdot {{y}^{-3/2}} \right)}^{12}}\;\;\; & 12. \;\;\;{{81}^{-1/2}}\;\;\; \\
13. \;\;\;{{\left( \frac{1}{16} \right)}^{3/4}}\;\;\; & 14. \;\;\;{{\left( 64{{a}^{3}} \right)}^{1/3}}\;\;\; & 15. \;\;\;{{\left( \frac{9}{4} \right)}^{-3/2}}\;\;\; \\
16. \;\;\;\frac{{{3}^{1/6}}\cdot {{9}^{2/3}}}{{{27}^{1/2}}}\;\;\; & 17. \;\;\;{{\left( 8{{a}^{3}}{{n}^{1/2}}{{p}^{-3/4}} \right)}^{2/3}}\;\;\; & 18. \;\;\;{{\left( {{x}^{mn/2}} \right)}^{4/nm}}\;\;\; \\
19. \;\;\;{{\left( \frac{{{x}^{6}}{{y}^{-3}}}{27} \right)}^{-2/3}}\;\;\; & 20. \;\;\;{{\left( \frac{{{3}^{x}}{{a}^{3x}}}{{{b}^{x/3}}} \right)}^{3/x}}\;\;\; & 21. \;\;\;{{\left( \frac{{{x}^{1/3}}{{x}^{4/3}}}{{{x}^{-1/3}}} \right)}^{2}}\;\;\; \\
22. \;\;\;{{\left( {{x}^{\tfrac{mn}{2}}} \right)}^{\tfrac{4}{nm}}}\;\;\; & 23. \;\;\;\frac{{{2}^{-5/6}}\cdot {{2}^{-2/3}}}{{{4}^{1/4}}}\;\;\; & 24. \;\;\;{{\left( \frac{{{x}^{-3}}}{8{{x}^{-6}}} \right)}^{2/3}}\;\;\; \\
25. \;\;\;{{\left( \frac{{{x}^{-3}}}{{{y}^{-6}}} \right)}^{-1/3}}\;\;\; & 26. \;\;\;\frac{{{x}^{1/2}}{{x}^{1/3}}{{y}^{4}}}{{{y}^{1/2}}{{y}^{3/2}}{{x}^{-1/6}}}\;\;\; & 27. \;\;\;{{8}^{-2/3}}-{{(2x)}^{0}}\cdot {{2}^{-2}}+\frac{1}{{{3}^{-2}}}\;\;\; \\
28. \;\;\;\frac{(x+y){{(x+y)}^{4}}}{{{(x+y)}^{1/2}}{{(x+y)}^{5/2}}}\;\;\; & 29. \;\;\;{{\left( \frac{81{{a}^{5}}{{b}^{4}}}{3{{a}^{2}}{{b}^{7}}} \right)}^{1/3}}\;\;\; & 30. \;\;\;\frac{{{m}^{1/2}}\cdot n}{{{\left( {{m}^{-1}}+1 \right)}^{-1/2}}{{\left( m+1 \right)}^{1/2}}}\;\;\; \\
31. \;\;\;\frac{{{m}^{1/3}}{{n}^{1/3}}{{\left( {{m}^{2}}{{n}^{2}} \right)}^{2/3}}}{{{(mn)}^{2/3}}}\;\;\; & 32. \;\;\;{{\left( {{4}^{-1/2}}+{{9}^{-1/2}} \right)}^{2}}\;\;\; & 33. \;\;\;\left( {{x}^{1/2}}+{{y}^{1/2}} \right)\left( {{x}^{1/2}}-{{y}^{1/2}} \right)\;\;\; \\
34. \;\;\;{{\left( {{x}^{1/2}}-{{y}^{1/2}} \right)}^{2}}\;\;\; & 35. \;\;\;{{\left( {{a}^{-1/2}}+{{b}^{-1/2}} \right)}^{2}}\;\;\; & 36. \;\;\;\frac{\left( {{a}^{1/2}}+{{b}^{1/2}} \right)\left( {{a}^{1/2}}-{{b}^{1/2}} \right)}{{{a}^{2}}-{{b}^{2}}}\;\;\;
\end{array}
Rewrite each of the following with fractional exponents and then simplify.
\begin{array}{l l}
37. \;\;\;\sqrt[3]{16{{t}^{2}}}\cdot \sqrt{8{{t}^{3}}}\;\;\; & 38. \;\;\;\sqrt[4]{25{{t}^{2}}}\cdot \sqrt{5t}\;\;\; & 39. \;\;\;\sqrt[4]{3K}\cdot \sqrt[3]{9{{K}^{2}}}\;\;\; \\
40. \;\;\;\sqrt[3]{\frac{{{x}^{2}}}{{{y}^{-3}}}}\;\;\; & 41. \;\;\;\sqrt[4]{\sqrt[3]{{{x}^{2}}}}\;\;\; & 42. \;\;\;\sqrt[3]{\sqrt[4]{{{x}^{12}}}}\;\;\; \\
43. \;\;\;\sqrt[3]{32{{t}^{2}}}\cdot \sqrt[6]{4{{t}^{2}}}\;\;\; & 44. \;\;\;\sqrt{2t}\cdot \sqrt[3]{4{{t}^{2}}}\cdot \sqrt[4]{8{{t}^{3}}}\;\;\; & 45. \;\;\;\sqrt{8{{t}^{3}}}\cdot \sqrt[3]{4{{t}^{2}}}\cdot \sqrt[4]{2t}\;\;\; \\
46. \;\;\;\frac{\sqrt{8{{t}^{3}}}}{\sqrt[3]{4{{t}^{2}}}}\;\;\; & 47. \;\;\;\frac{\sqrt{27{{t}^{3}}}}{\sqrt[4]{3t}}\;\;\; & 48. \;\;\;\frac{\sqrt{32{{s}^{3}}{{t}^{5}}}}{\sqrt[3]{2{{s}^{2}}{{t}^{4}}}}\;\;\;
\end{array}
Identify each of the following as TRUE or FALSE.
\begin{array}{l l}
49. \;\;\;\frac{{{x}^{1/4}}{{y}^{2/3}}}{{{x}^{3/4}}{{y}^{1/3}}}=\frac{\sqrt[3]{y}}{\sqrt{x}}\;\;\; & 50. \;\;\;{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{1/2}}=x+y\;\;\; \\
51. If \;\;\;x>0\;\;\;, then \;\;\;\sqrt[3]{\sqrt{x}}=\sqrt{\sqrt[3]{x}}\;\;\;. & 52. \;\;\;{{x}^{-2/3}}+4{{x}^{-5/3}}={{x}^{-5/3}}\left( x+4 \right)\;\;\; \\
53. \;\;\;{{\left( {{x}^{1/2}}+{{y}^{1/2}} \right)}^{2}}=x+y\;\;\; & 54. \;\;\;\sqrt{x}\cdot \sqrt[4]{{{x}^{3}}}\cdot \sqrt[6]{{{x}^{5}}}\cdot \sqrt[12]{{{x}^{11}}}={{x}^{3}}\;\;\;
\end{array}