Question

The value of${512}^{\frac{-2}{9}}$ is

Answer 1

Find the value of${512}^{\frac{-2}{9}}$

${512}^{\frac{-2}{9}}$

This expression can be written as,

$\begin{array}{rcl}{512}^{\frac{-2}{9}}& =& {\left(2^9\right)}^{\frac{-2}{9}}\end{array}$

$\begin{array}{rcl}& & \mathrm{By}\mathrm{using}\mathrm{law}\mathrm{of}\mathrm{exponent},\end{array}$

$\begin{array}{rcl}{\left({\mathrm{a}}^{\mathrm{m}}\right)}^{\mathrm{n}}& =& {\mathrm{a}}^{\mathrm{mn}}\\ \mathrm{then}{512}^{\frac{-2}{9}}& =& {\left(2^9\right)}^{\frac{-2}{9}}\\ & =& 2^{9\times \frac{-2}{9}}\\ & =& 2^{-2}\\ & =& \frac{1}{2^2}\left[\because {\mathrm{a}}^{-\mathrm{m}}=\frac{1}{{\mathrm{a}}^{\mathrm{m}}}\right]\\ & =& \frac{1}{4}\\ \therefore {512}^{\frac{-2}{9}}& =& \frac{1}{4}\end{array}$