# How do you solve x^ (-2/3) = 9?

May 11, 2018

raise both sides to the $- \frac{3}{2}$ power.

Then you have:
${x}^{\left(- \frac{2}{3}\right) \left(- \frac{3}{2}\right)} = {9}^{- \frac{3}{2}}$

$x = {9}^{- \frac{3}{2}} = {\left({3}^{2}\right)}^{- \frac{3}{2}}$

$x = {3}^{-} 3$

$x = \frac{1}{27}$

May 11, 2018

A trick using logs.

$x = \frac{1}{27}$

#### Explanation:

Given: ${x}^{- \frac{2}{3}} = 9$

Take logs of both sides:

$\ln \left({x}^{- \frac{2}{3}}\right)$=ln(9)#

$- \frac{2}{3} \ln \left(x\right) = \ln \left(9\right)$

$\ln \left(x\right) = - \frac{3}{2} \ln \left(9\right)$

$x = {\ln}^{- 1} \left[- \frac{3}{2} \ln \left(9\right)\right]$

$x = 0.037 \textcolor{w h i t e}{.} \overline{037}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$1000 x = 37.037 \overline{037}$
$\underline{\textcolor{w h i t e}{1000} x = \textcolor{w h i t e}{0} 0.037 \overline{037} \leftarrow \text{ Subtract}}$
$\textcolor{w h i t e}{\text{d}} 999 x = 37$

$x = \frac{37}{999} = \frac{1}{27}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Check}}$

${x}^{- \frac{2}{3}} = \frac{1}{{x}^{\frac{2}{3}}}$

Consider ${x}^{\frac{2}{3}} \to {\left(\frac{1}{27}\right)}^{\frac{2}{3}} = 0.11 \overline{1}$

$\frac{1}{{x}^{\frac{2}{3}}} = \frac{1}{0.11 \overline{1}} = 9 \textcolor{red}{\leftarrow \text{ As required}}$