Question

Which among the following would be equal to ${81}^{\frac{2}{3}}?$

• A. Cube root of $81$
• B. Square root of $81$
• C. Square root of $9^4$
• D. Cube root of $9^4$

Answer 1

The correct option is D Cube root of $9^4$

$\underset{–}{\text{Need to simplify:}}{81}^{\frac{2}{3}}$

$81$ can be expressed as: $81=9\times 9=9^2$

$\therefore {81}^{\frac{2}{3}}=(9^2{)}^{\frac{2}{3}}$

$=(9^{2\times 2}{)}^{\frac{1}{3}}$ $[\because (a^m{)}^{\frac{p}{q}}=(a^{m\times p}{)}^{\frac{1}{q}}]$

$=(9^4{)}^{\frac{1}{3}}$

$\Rightarrow \underset{–}{{81}^{\frac{2}{3}}=\sqrt[3]{39^4}}$

From the above equation, it is clear that ${81}^{\frac{2}{3}}$ is equal to the cube root of $9^4$.

Therefore, option (d.) is the correct answer.