Mathematics » Exponents and Surds » Rational Exponents

Rational Exponents

Rational Exponents

We can also apply the exponent laws to expressions with rational exponents.

Example

Question

Simplify:

\[2{x}^{\frac{1}{2}}\times 4{x}^{-\cfrac{1}{2}}\]

\begin{align*} 2{x}^{\frac{1}{2}} \times 4{x}^{-\cfrac{1}{2}} & = 8{x}^{\frac{1}{2} – \cfrac{1}{2}}\\ & = 8{x}^{0} \\ & = 8(1) \\ & = 8 \end{align*}

Example

Question

Simplify:

\[{(\text{0.008})}^{\frac{1}{3}}\]

Write as a fraction and simplify

\begin{align*} {(\text{0.008})}^{\frac{1}{3}} & = {\left(\cfrac{8}{\text{1 000}}\right)}^{\frac{1}{3}} \\ & = {\left(\cfrac{1}{125}\right)}^{\frac{1}{3}} \\ & = {\left(\cfrac{1}{5^{3}}\right)}^{\frac{1}{3}} \\ & = \cfrac{{1}^{\frac{1}{3}}}{5^{(3 \cdot \cfrac{1}{3})}} \\ & = \cfrac{1}{5} \end{align*}

Extension: the following video provides a summary of all the exponent rules and rational exponents.

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