Answer
$243 \,mm$
Work Step by Step
If the length of one side is $L$, then the volume
$V=L^{3}=27^{5}\,mm^{3}$
$\implies L= (27^{5}\,mm^{3})^{1/3}=27^{\frac{5}{3}}\,mm$
$27$ can be written as $3^{3}$.
$\implies L=(3^{3})^{\frac{5}{3}}\,mm=3^{3\cdot\frac{5}{3}}\,mm=3^{5}\,mm$
$=243\,mm$
