We think you are located in South Africa. Is this correct?

Rational Exponents

2.3 Rational exponents (EMAV)

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

According to CAPS, the rational exponent law is introduced in Grade 11 but you may choose to introduce learners to the rational exponent law \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) at this stage.

Worked example 6: Simplifying rational exponents

Simplify:

\[2{x}^{\frac{1}{2}}\times 4{x}^{-\frac{1}{2}}\]
\begin{align*} 2{x}^{\frac{1}{2}} \times 4{x}^{-\frac{1}{2}} & = 8{x}^{\frac{1}{2} - \frac{1}{2}}\\ & = 8{x}^{0} \\ & = 8\left(1\right) \\ & = 8 \end{align*}

Worked example 7: Simplifying rational exponents

Simplify:

\[{\left(\text{0,008}\right)}^{\frac{1}{3}}\]

Write as a fraction and simplify

\begin{align*} {\left(\text{0,008}\right)}^{\frac{1}{3}} & = {\left(\frac{8}{\text{1 000}}\right)}^{\frac{1}{3}} \\ & = {\left(\frac{1}{125}\right)}^{\frac{1}{3}} \\ & = {\left(\frac{1}{5^{3}}\right)}^{\frac{1}{3}} \\ & = \frac{{1}^{\frac{1}{3}}}{5^{\left(3 \cdot \frac{1}{3}\right)}} \\ & = \frac{1}{5} \end{align*}

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

Video: 2F2V

Do you need more Practice?

Siyavula Practice gives you access to unlimited questions with answers that help you learn. Practise anywhere, anytime, and on any device!

Sign up to practise now

Exercise 2.2

Simplify without using a calculator:

\(t^{\frac{1}{4}} \times 3t^{\frac{7}{4}}\)

\begin{align*} t^{\frac{1}{4}} \times 3t^{\frac{7}{4}} & = 3t^{\frac{1}{4} + \frac{7}{4}} \\ & = 3t^{\frac{8}{4}} \\ & = 3t^{2} \end{align*}

\(\dfrac{16x^{2}}{\left(4x^{2}\right)^{\frac{1}{2}}}\)

\begin{align*} \frac{16x^{2}}{\left(4x^{2}\right)^{\frac{1}{2}}} & = \frac{4^{2}x^{2}}{4^{\frac{1}{2}}x^{(2)\left(\frac{1}{2}\right)}} \\ & = \frac{4^{2}x^{2}}{4^{\frac{1}{2}}x} \\ & = 4^{2 - \frac{1}{2}} \cdot x^{2 - 1} \\ & = \left(2^{2}\right)^{\frac{3}{2}}x \\ & = 2^{3}x \\ & = 8x \end{align*}

\(\left(\text{0,25}\right)^{\frac{1}{2}}\)

\begin{align*} \left(\text{0,25}\right)^{\frac{1}{2}} & = \left(\dfrac{1}{4}\right)^{\frac{1}{2}} \\ & = \left(\dfrac{1}{2^{2}}\right)^{\frac{1}{2}} \\ & = \left(2^{-2}\right)^{\frac{1}{2}} \\ & = 2^{-1} \\ & = \dfrac{1}{2} \end{align*}

\(\left(27\right)^{-\frac{1}{3}}\)

\begin{align*} \left(27\right)^{-\frac{1}{3}} & = \left(3^{3}\right)^{-\frac{1}{3}} \\ & = 3^{-1} \\ & = \dfrac{1}{3} \end{align*}

\(\left(3p^{2}\right)^{\frac{1}{2}} \times \left(3p^{4}\right)^{\frac{1}{2}}\)

\begin{align*} \left(3p^{2}\right)^{\frac{1}{2}} \times \left(3p^{4}\right)^{\frac{1}{2}} & = 3^{\frac{1}{2}}p \times 3^{\frac{1}{2}}p^{2} \\ & = 3^{\frac{1}{2} + \frac{1}{2}} \times p^{1 + 2} \\ & = 3p^{3} \end{align*}

\(\text{12} {\left( a^\text{4}b^\text{8} \right)}^ {\frac{\text{1}}{\text{2}}} \times {\left( \text{512}a^\text{3}b^\text{3} \right)}^ {\frac{\text{1}}{\text{3}}}\)

\begin{align*} \text{12} {\left( a^\text{4}b^\text{8} \right)}^ {\frac{\text{1}}{\text{2}}} \times {\left( \text{512}a^\text{3}b^\text{3} \right)}^ {\frac{\text{1}}{\text{3}}} &= \text{12} a^{\frac{\text{4}}{\text{2}}}b^ {\frac{\text{8}}{\text{2}}} \times (\text{512})^{\frac{\text{1}}{\text{3}}}a^{\frac{\text{3}}{\text{3}}}b^{\frac{\text{3}}{\text{3}}} \\ &= \text{12} a^{\text{2}}b^{\text{4}} \times \left( \text{8}^{\text{3}} \right) ^{\frac{\text{1}}{\text{3}}}a^{\text{1}}b^{\text{1}} \\ &= \text{12} a^{\text{2}}b^{\text{4}} \times \text{8}a^{\text{1}}b^{\text{1}} \\ &= \text{96} a^{\text{3}} b^{\text{5}} \end{align*}

\(\left((-2)^4a^6b^2\right)^{\frac{1}{2}}\)
\begin{align*} \left((-2)^4a^6b^2\right)^{\frac{1}{2}} & = (-2)^2(a^3b) \\ & = 4a^3b \end{align*}
\(\left(a^{-2}b^6\right)^{\frac{1}{2}}\)
\begin{align*} \left(a^{-2}b^6\right)^{\frac{1}{2}} & = a^{-1}b^3 \\ & = \frac{b^{3}}{a} \end{align*}
\(\left(16x^{12}b^6\right)^{\frac{1}{3}}\)
\begin{align*} \left(16x^{12}b^6\right)^{\frac{1}{3}} & = \left((8 \times 2) x^{12}b^{6}\right)^{\frac{1}{3}} \\ & = 2\cdot 2^{\frac{1}{3}}a^{4}b^{2} \end{align*}