ANSWER
\({(243)}^{ \frac{2}{5} } = 9\)
EXPLANATION
We want to simplify the exponential expression,
\({(243)}^{ \frac{2}{5} }\)
We use the laws of indices to simplify the expression.
We rewrite the above expression to obtain,
\({(243)}^{ \frac{2}{5} } = {(243)}^{ \frac{1}{5} \times 2 }\)
Recall that,
\({a}^{mn} = ( {a}^{m} ) ^{n}\)
When we apply the above property, we get
\({(243)}^{ \frac{2}{5} } = {( {(243)}^{ \frac{1}{5} } )}^{2}\)
Now we need to write
\(243\)
as a certain number to the exponent of 5.
In order words,
\(243 = 3 \times 3 \times 3 \times 3 \times 3 = {3}^{5}\)
This implies that,
\({(243)}^{ \frac{2}{5} } = {( { {3}^{5} }^{ \times \frac{1}{5} } )}^{2}\)
We further simplify to get,
\({(243)}^{ \frac{2}{5} } = 3^{2}\)
This will finally evaluate to,
\({(243)}^{ \frac{2}{5} } = 9\)
