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 apply as for integer exponents.

Eliminate the brackets and write the answer as one fraction.
$\left(c^{\frac{1}{5}} \cdot x \cdot d^{-3}\right)^{3}$
Simplify your answer as much as possible and write it without any fractional powers and any negative exponents.

$\dfrac{ \sqrt[5]{c^3} \cdot x^{3}}{d^{9}}$

$\begin{array}{rcl} \left(c^{\frac{1}{5}} \cdot x \cdot d^{-3}\right)^{3} &=& \left(c^{\frac{1}{5}}\right)^{3} \cdot x^{3} \cdot \left(d^{-3}\right)^{3} \\ &&\phantom{xxx}\blue{\text{rule } \left(a \cdot b \right)^{n} = a^{n} \cdot b^{n}} \\ &=& c^{\frac{1}{5} \cdot 3} \cdot x^{3} \cdot d^{-3 \cdot 3} \\ &&\phantom{xxx}\blue{\text{rule } \left(a^{n}\right)^{m} = a^{n \cdot m}} \\ &=& c^{{{3}\over{5}}} \cdot x^{3} \cdot d^{-9} \\ &&\phantom{xxx}\blue{\text{exponents simplified}}\\ &=& \dfrac{ \sqrt[5]{c^3} \cdot x^{3}}{d^{9}} \\ &&\phantom{xxx}\blue{\text{negative exponent and fractional power eliminated with rules }} \\&&\phantom{xxx}\blue{a^{-n}=\frac{1}{a^n} \text{ and } a^{\frac{m}{n}}=\sqrt[n]{a^m}} \end{array}$

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