Find the Inverse t=a^(3/2)

Rewrite the equation as ${\displaystyle y^{\frac{3}{2}}=a}$.

Raise each side of the equation to the power of ${\displaystyle \frac{2}{3}}$ to eliminate the fractional exponent on the left side.

Simplify the left side.

To verify the inverse, check if ${\displaystyle f^{-1}\left(f\left(a\right)\right)=a}$ and ${\displaystyle f\left(f^{-1}\left(a\right)\right)=a}$.

Evaluate ${\displaystyle f^{-1}\left(f\left(a\right)\right)}$.

Evaluate ${\displaystyle f\left(f^{-1}\left(a\right)\right)}$.

Since ${\displaystyle f^{-1}\left(f\left(a\right)\right)=a}$ and ${\displaystyle f\left(f^{-1}\left(a\right)\right)=a}$, then ${\displaystyle f^{-1}\left(a\right)=a^{\frac{2}{3}}}$ is the inverse of ${\displaystyle f\left(a\right)=a^{\frac{3}{2}}}$.