Question

Evaluate: ${\left(27\right)}^{-\frac{1}{3}}\times {\left(27\right)}^{-\frac{1}{3}}\left[{27}^{\frac{1}{3}}-{27}^{\frac{2}{3}}\right]$

Answer 1

Given ${\left(27\right)}^{-\frac{1}{3}}\times {\left(27\right)}^{-\frac{1}{3}}\left[{27}^{\frac{1}{3}}-{27}^{\frac{2}{3}}\right]$
$=(3^3{)}^{-\frac{1}{3}}\times (3^3{)}^{-\frac{1}{3}}\left[\right(3^3{)}^{\frac{1}{3}}-\left(3^3{)}^{\frac{2}{3}}\right]$
$=3^{-(3\times \frac{1}{3})}\times 3^{-(3\times \frac{1}{3})}[3^{(3\times \frac{1}{3})}-3^{(3\times \frac{2}{3})}]$ [Since, $(a^m{)}^n=a^{(m\times n)}$]
$=3^{-1}\times 3^{-1}[3^1-3^2]$
$=\frac{1}{3}\times \frac{1}{3}\times (3-9)$
$=\frac{1}{9}\times (-6)$
$=-\frac{2}{3}$
Hence, the simplified value of the given expression is $-\frac{2}{3}$.