Question

How do you simplify ${\left(\frac{2}{3}\right)}^{-3}$?

Answer 1

Step $1$: Find inverse:

Using the law of exponents,${\left(\frac{2}{3}\right)}^{-3}$can be written as ${\left[{\left(\frac{2}{3}\right)}^3\right]}^{-1}$.

As we know, $x^{-1}=\frac{1}{x}$

Thus ${\left(\frac{2}{3}\right)}^{-3}$can be written as $\frac{1}{{\left({\displaystyle \frac{2}{3}}\right)}^3}$.

Step $2$: Simplify the required term:

${\left(\frac{2}{3}\right)}^{-3}=\frac{1}{{\left({\displaystyle \frac{2}{3}}\right)}^3}$

$$\begin{gathered} \Rightarrow {\left(\frac{2}{3}\right)}^{-3}=\frac{3^3}{2^3} \\ \Rightarrow {\left(\frac{2}{3}\right)}^{-3}=\frac{27}{8} \end{gathered}$$

Hence, ${\left(\frac{2}{3}\right)}^{-3}$ can be simplified as $\frac{27}{8}$.