When simplifying radicals that use fractional exponents, the numerator on the exponent is divided by the denominator. All radicals can be shown as having an equivalent fractional exponent. For example:

[latex]\sqrt{x}=x^{\frac{1}{2}}\hspace{0.25in} \sqrt[3]{x}=x^{\frac{1}{3}}\hspace{0.25in} \sqrt[4]{x}=x^{\frac{1}{4}}\hspace{0.25in} \sqrt[5]{x}=x^{\frac{1}{5}}[/latex]

Radicals having some exponent value inside the radical can also be written as a fractional exponent. For example:

[latex]\sqrt{x^3}=x^{\frac{3}{2}}\hspace{0.25in} \sqrt[3]{x^2}=x^{\frac{2}{3}}\hspace{0.25in} \sqrt[4]{x^5}=x^{\frac{5}{4}}\hspace{0.25in} \sqrt[5]{x^9}=x^{\frac{9}{5}}[/latex]

The general form that radicals having exponents take is:

[latex]x^{\frac{b}{a}}=\sqrt[a]{x^b}\text{ or }(\sqrt[a]{x})^b[/latex]

Should the reciprocal of a radical having an exponent, it would look as follows:

[latex]x^{-\frac{b}{a}}=\dfrac{1}{\sqrt[a]{x^b}}\text{ or }\dfrac{1}{(\sqrt[a]{x})^b}[/latex]

In both cases shown above, the power of the radical is [latex]b[/latex] and the root of the radical is [latex]a[/latex]. These are the two forms that a radical having an exponent is commonly written in. It is convenient to work with a radical containing an exponent in one of these two forms.

Once the radical having an exponent is converted into a pure fractional exponent, then the following rules can be used.

Properties of Exponents

[latex]\begin{array}{ccc} a^ma^n=a^{m+n}\hspace{0.25in} &(ab)^m=a^mb^m\hspace{0.25in} &a^{-m}=\dfrac{1}{a^m} \\ \\ \dfrac{a^m}{a^n}=a^{m-n}&\left(\dfrac{a}{b}\right)=\dfrac{a^m}{b^m}&\dfrac{1}{a^{-m}}=a^m \\ \\ (a^m)^n=a^{mn}&a^0=1&\left(\dfrac{a}{b}\right)^{-m}=\dfrac{b^m}{a^m} \end{array}[/latex]

Questions

Write each of the following fractional exponents in radical form.

1. [latex]m^{\frac{3}{5}}[/latex]
2. [latex](10r)^{-\frac{3}{4}}[/latex]
3. [latex](7x)^{\frac{3}{2}}[/latex]
4. [latex](6b)^{-\frac{4}{3}}[/latex]
5. [latex](2x+3)^{-\frac{3}{2}}[/latex]
6. [latex](x-3y)^{\frac{3}{4}}[/latex]

Write each of the following radicals in exponential form.

7. [latex]\sqrt[3]{5}[/latex]
8. [latex]\sqrt[5]{2^3}[/latex]
9. [latex]\sqrt[3]{ab^5}[/latex]
10. [latex]\sqrt[5]{x^3}[/latex]
11. [latex]\sqrt[3]{(a+5)^2}[/latex]
12. [latex]\sqrt[5]{(a-2)^3}[/latex]

Evaluate the following.

13. [latex]8^{\frac{2}{3}}[/latex]
14. [latex]16^{\frac{1}{4}}[/latex]
15. [latex]\sqrt[3]{4^6}[/latex]
16. [latex]\sqrt[5]{32^2}[/latex]

Simplify. Your answer should only contain positive exponents.

17. [latex](xy^{\frac{1}{3}})(xy^{\frac{2}{3}})[/latex]
18. [latex](4v^{\frac{2}{3}})(v^{-1})[/latex]
19. [latex](a^{\frac{1}{2}}b^{\frac{1}{2}})^{-1}[/latex]
20. [latex](x^{\frac{5}{3}}y^{-2})^0[/latex]
21. [latex]\dfrac{a^2b^0}{3a^4}[/latex]
22. [latex]\dfrac{2x^{\frac{1}{2}}y^{\frac{1}{3}}}{2x^{\frac{4}{3}}y^{\frac{7}{4}}}[/latex]
23. [latex]\dfrac{a^{\frac{3}{4}}b^{-1}b^{\frac{7}{4}}}{3b^{-1}}[/latex]
24. [latex]\dfrac{2x^{-2}y^{\frac{5}{3}}}{x^{-\frac{5}{4}}y^{-\frac{5}{3}}xy^{\frac{1}{2}}}[/latex]
25. [latex]\dfrac{3y^{-\frac{5}{4}}}{y^{-1}2y^{-\frac{1}{3}}}[/latex]
26. [latex]\dfrac{ab^{\frac{1}{3}}2b^{-\frac{5}{4}}}{4a^{-\frac{1}{2}}b^{-\frac{2}{3}}}[/latex]

Answer Key 9.6