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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:
\(\sqrt{x}=x^{\frac{1}{2}}\hspace{0.25in} \sqrt[3]{x}=x^{\frac{1}{3}}\hspace{0.25in} \sqrt[4]{x}=x^{\frac{1}{4}}\hspace{0.25in} \sqrt[5]{x}=x^{\frac{1}{5}}\)
Radicals having some exponent value inside the radical can also be written as a fractional exponent. For example:
\(\sqrt{x^3}=x^{\frac{3}{2}}\hspace{0.25in} \sqrt[3]{x^2}=x^{\frac{2}{3}}\hspace{0.25in} \sqrt[4]{x^5}=x^{\frac{5}{4}}\hspace{0.25in} \sqrt[5]{x^9}=x^{\frac{9}{5}}\)
The general form that radicals having exponents take is:
\(x^{\frac{b}{a}}=\sqrt[a]{x^b}\text{ or }(\sqrt[a]{x})^b\)
Should the reciprocal of a radical having an exponent, it would look as follows:
\(x^{-\frac{b}{a}}=\dfrac{1}{\sqrt[a]{x^b}}\text{ or }\dfrac{1}{(\sqrt[a]{x})^b}\)
In both cases shown above, the power of the radical is \(b\) and the root of the radical is \(a\). 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.
Example 1
Evaluate \(27^{-\frac{4}{3}}\).
Converting to a radical form:
\(\dfrac{1}{\sqrt[3]{27^4}}\text{ or }\dfrac{1}{(\sqrt[3]{27})^4}\)
First, the cube root of 27 will reduce to 3, which leaves:
\(\dfrac{1}{3^4}\text{ or }\dfrac{1}{81}\)
Once the radical having an exponent is converted into a pure fractional exponent, then the following rules can be used.
Properties of Exponents
\(\begin{array}{ccc}
a^ma^n=a^{m+n}\hspace{0.25in} &(ab)^m=a^mb^m\hspace{0.25in} &a^{-m}=\dfrac{1}{a^m} \\ \\
\dfrac{a^m}{a^n}=a^{m-n}&\left(\dfrac{a}{b}\right)=\dfrac{a^m}{b^m}&\dfrac{1}{a^{-m}}=a^m \\ \\
(a^m)^n=a^{mn}&a^0=1&\left(\dfrac{a}{b}\right)^{-m}=\dfrac{b^m}{a^m}
\end{array}\)
Example 2
Simplify \((x^2y^{\frac{4}{3}})(x^{-1}y^\frac{2}{3})\).
First, you need to separate the different variables:
\((x^2y^{\frac{4}{3}})(x^{-1}y^\frac{2}{3})\) becomes \(x^2\cdot x^{-1}\cdot y^{\frac{4}{3}}\cdot y^{\frac{2}{3}}\)
Combining the exponents yields:
\(x^{2 – 1}\cdot y^{\frac{4}{3}+\frac{2}{3}}\)
Which results in:
\(x^1\cdot y^{\frac{6}{3}}\)
Which simplifies to:
\(xy^2\)
Example 3
Simplify \(\dfrac{ab^{\frac{2}{3}}3b^{-\frac{5}{3}}}{5a^{-\frac{3}{2}}b^{-\frac{4}{3}}}\).
First, separate the different variables:
\(\dfrac{ab^{\frac{2}{3}}3b^{-\frac{5}{3}}}{5a^{-\frac{3}{2}}b^{-\frac{4}{3}}}\) becomes \(3\cdot 5^{-1}\cdot a \cdot a^{\frac{3}{2}}\cdot b^{\frac{2}{3}}\cdot b^{-\frac{5}{3}}\cdot b^{\frac{4}{3}}\)[1]
Combining the exponents yields:
\(3\cdot 5^{-1}\cdot a^{1+\frac{3}{2}}\cdot b^{\frac{2}{3}-\frac{5}{3}+\frac{4}{3}}\)
Which gives:
\(3\cdot 5^{-1}\cdot a^{\frac{5}{2}}\cdot b^{\frac{1}{3}}\)
Which simplifies to:
\(\dfrac{3\cdot a^{\frac{5}{2}}\cdot b^{\frac{1}{3}}}{5}\)
Questions
Write each of the following fractional exponents in radical form.
- \(m^{\frac{3}{5}}\)
- \((10r)^{-\frac{3}{4}}\)
- \((7x)^{\frac{3}{2}}\)
- \((6b)^{-\frac{4}{3}}\)
- \((2x+3)^{-\frac{3}{2}}\)
- \((x-3y)^{\frac{3}{4}}\)
Write each of the following radicals in exponential form.
- \(\sqrt[3]{5}\)
- \(\sqrt[5]{2^3}\)
- \(\sqrt[3]{ab^5}\)
- \(\sqrt[5]{x^3}\)
- \(\sqrt[3]{(a+5)^2}\)
- \(\sqrt[5]{(a-2)^3}\)
Evaluate the following.
- \(8^{\frac{2}{3}}\)
- \(16^{\frac{1}{4}}\)
- \(\sqrt[3]{4^6}\)
- \(\sqrt[5]{32^2}\)
Simplify. Your answer should only contain positive exponents.
- \((xy^{\frac{1}{3}})(xy^{\frac{2}{3}})\)
- \((4v^{\frac{2}{3}})(v^{-1})\)
- \((a^{\frac{1}{2}}b^{\frac{1}{2}})^{-1}\)
- \((x^{\frac{5}{3}}y^{-2})^0\)
- \(\dfrac{a^2b^0}{3a^4}\)
- \(\dfrac{2x^{\frac{1}{2}}y^{\frac{1}{3}}}{2x^{\frac{4}{3}}y^{\frac{7}{4}}}\)
- \(\dfrac{a^{\frac{3}{4}}b^{-1}b^{\frac{7}{4}}}{3b^{-1}}\)
- \(\dfrac{2x^{-2}y^{\frac{5}{3}}}{x^{-\frac{5}{4}}y^{-\frac{5}{3}}xy^{\frac{1}{2}}}\)
- \(\dfrac{3y^{-\frac{5}{4}}}{y^{-1}2y^{-\frac{1}{3}}}\)
- \(\dfrac{ab^{\frac{1}{3}}2b^{-\frac{5}{4}}}{4a^{-\frac{1}{2}}b^{-\frac{2}{3}}}\)
Answers to odd questions
1. \(\sqrt[5]{m^3}\)
3. \(\sqrt{(7x)^3}\)
5. \(\dfrac{1}{\sqrt{(2x+3)^3}}\)
7. \(5^{\frac{1}{3}}\)
9. \((ab^5)^{\frac{1}{3}}\) or \(a^{\frac{1}{3}}b^{\frac{5}{3}}\)
11. \((a+5)^{\frac{2}{3}}\)
13. \(8^{\frac{2}{3}}\Rightarrow (2^3)^{\frac{2}{3}}\Rightarrow 2^2\text{ or }4\)
15. \(\sqrt[3]{4^6}\Rightarrow (2^2)^{\frac{6}{3}}\Rightarrow 2^4\text{ or }16\)
17. \(x^2y^{\frac{1}{3}+\frac{2}{3}}\Rightarrow x^2y\)
19. \(a^{-\frac{1}{2}}b^{-\frac{1}{2}}\Rightarrow \dfrac{1}{a^{\frac{1}{2}}b^{\frac{1}{2}}}}\)
21. \(\dfrac{\cancel{a^2}\cancel{b^0}1}{3\cancel{a^4}a^2}\Rightarrow \dfrac{1}{3a^2}\)
23. \(\dfrac{a^{\frac{3}{4}}\cancel{b^{-1}}b^{\frac{7}{4}}}{3\cancel{b^{-1}}}\Rightarrow \dfrac{a^{\frac{3}{4}}b^{\frac{7}{4}}}{3}\)
25. \(\dfrac{3}{2}y^{-\frac{5}{4}- -1 – -\frac{1}{3}} \Rightarrow \dfrac{3}{2}y^{\frac{1}{12}}\)
- When we divide by an exponent, we subtract powers. ↵
