Calculating with fractional exponents

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

\[\begin{array}{rcl}\blue{x}^{\frac{\purple{3}}{\orange{2}}} &=& \sqrt{\blue{x^\purple{3}}}\\ \\ \blue{x}^{\frac{\purple{3}}{\orange{5}}}&=&\sqrt[\orange{5}]{\blue{x^\purple{3}}}\end{array}\]

This rule can be reduced from the first rule.

\[\begin{array}{rcl}\sqrt[\orange n]{\blue a^\purple m}&=& \left(\blue a^\purple m\right)^{\frac{1}{\orange n}}\\&& \phantom{xxxxx}\blue{\text{rule }\sqrt[n]{a}=\blue a^{\frac{1}{n}}}\\&=&\blue a^{\frac{\purple m}{\orange n}}\\&& \phantom{xxxxx}\blue{\text{rule for exponents }\left(a^m\right)^n=a^{m \cdot n}} \end{array}\]

For fractional exponents the same rules as for integer exponents apply.

Eliminate the brackets and write the answer as one fraction.
\[\left(a^{\frac{1}{2}} \cdot z \cdot p^{-6}\right)^{2}\]
Simplify your answer as much as possible and write it without any fractional powers and without any negative exponents.

#\dfrac{a \cdot z^{2}}{p^{12}}#

#\begin{array}{rcl}
\left(a^{\frac{1}{2}} \cdot z \cdot p^{-6}\right)^{2} &=& \left(a^{\frac{1}{2}}\right)^{2} \cdot z^{2} \cdot \left(p^{-6}\right)^{2} \\ &&\phantom{xxx}\blue{\text{rule } \left(a \cdot b \right)^{n} = a^{n} \cdot b^{n}} \\
&=& a^{\frac{1}{2} \cdot 2} \cdot z^{2} \cdot p^{-6 \cdot 2} \\ &&\phantom{xxx}\blue{\text{rule } \left(a^{n}\right)^{m} = a^{n \cdot m}} \\
&=& a^{1} \cdot z^{2} \cdot p^{-12}
\\ &&\phantom{xxx}\blue{\text{exponents simplified}}\\
&=& \dfrac{a \cdot z^{2}}{p^{12}} \\ &&\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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