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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

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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*}