Algebra: Calculating with exponents and roots
Calculating with fractional exponents
A fractional exponent is a power in which the exponent can be written as a fraction. A root can be written as a fractional exponent.
For #\blue a\geq 0# and integer #\orange n \geq 2# we have: \[\blue a^{\frac{1}{\orange n}}=\sqrt[\orange n]{\blue a}\] |
Examples \[\begin{array}{rcl}\blue{x}^{\frac{1}{\orange{2}}}&=& \sqrt{\blue{x}}\\ \\ \blue{x}^{\frac{1}{\orange{5}}}&=&\sqrt[\orange{5}]{\blue{x}}\end{array}\] |
For #\blue a \geq 0# and integers #\orange n, \purple m \geq 2# we have: \[\blue a^{\frac{\purple m}{\orange n}}=\sqrt[\orange n]{\blue a^\purple m}\] |
Examples \[\begin{array}{rcl}\blue{x}^{\frac{\purple{3}}{\orange{2}}} &=& \sqrt{\blue{x^\purple{3}}}\\ \\ \blue{x}^{\frac{\purple{3}}{\orange{5}}}&=&\sqrt[\orange{5}]{\blue{x^\purple{3}}}\end{array}\] |
For fractional exponents the same rules as for integer exponents apply.
#\begin{array}{rcl}
\left(c^{\frac{1}{6}} \cdot f \cdot b^{-2}\right)^{4} &=& \left(c^{\frac{1}{6}}\right)^{4} \cdot f^{4} \cdot \left(b^{-2}\right)^{4} \\ &&\phantom{xxx}\blue{\text{rule } \left(a \cdot b \right)^{n} = a^{n} \cdot b^{n}} \\
&=& c^{\frac{1}{6} \cdot 4} \cdot f^{4} \cdot b^{-2 \cdot 4} \\ &&\phantom{xxx}\blue{\text{rule } \left(a^{n}\right)^{m} = a^{n \cdot m}} \\
&=& c^{{{2}\over{3}}} \cdot f^{4} \cdot b^{-8}
\\ &&\phantom{xxx}\blue{\text{exponents simplified}}\\
&=& \dfrac{ \sqrt[3]{c^2} \cdot f^{4}}{b^{8}} \\ &&\phantom{xxx}\blue{\text{negative exponent and fractional power eliminated with rules }} \\&&\phantom{xxx}\blue{a^{-n}=\frac{1}{a^n} \text{ and } a^{\frac{m}{n}}=\sqrt[n]{a^m}}
\end{array}#
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