Question

How do you simplify $4$ to the power of $\frac{2}{3}$?

Answer 1

Step-1: Use the Fractional Exponent rule:

Given that, $4^{\frac{2}{3}}$

The fractional Exponent rule is $a^{\frac{m}{n}}=\sqrt[n]{a^m}$. Where $a$ is the base integer, $m$ is the numerator and $n$ is the denominator.

Substitute $a=4,m=2\mathrm{and}\mathrm{n}=3$ in the above rule:

$\begin{array}{rcl}& \Rightarrow & a^{\frac{m}{n}}=\sqrt[n]{a^m}\\ & \Rightarrow & 4^{\frac{2}{3}}=\sqrt[3]{4^2}\\ & =& \sqrt[3]{16}\\ & =& \sqrt[3]{8\left(2\right)}[\mathrm{Factor}8\mathrm{out}\mathrm{of}16]\\ \because & =& \sqrt[3]{2^3\times 2}[\mathrm{Rewrite}8\mathrm{as}{2}^3]\end{array}$

Step-2: Apply the radical rule:

Use the radical rule $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$:
$\begin{array}{rcl}& \Rightarrow & \sqrt[3]{2^3\times 2}=\sqrt[3]{2^3}\sqrt[3]{2}\\ & =& 2\sqrt[3]{2}[\therefore \sqrt[n]{a^n}=a]\end{array}$

Hence, the simplification of ${\mathbf{4}}^{\frac{\mathbf{2}}{\mathbf{3}}}$is $\mathbf{2}\sqrt[\mathbf{3}]{\mathbf{2}}$.