Solve and check the given equation with rational expo \( x^{3 / 2}=27 \)

To solve the equation \( x^{3/2} = 27 \), first, raise both sides to the power of \( \frac{2}{3} \) to isolate \( x \): \[ x = (27)^{\frac{2}{3}}. \] Calculating \( 27^{\frac{1}{3}} \) gives us \( 3 \), since \( 3^3 = 27 \). Now square it to find \( x \): \[ x = 3^2 = 9. \] To check the solution, substitute \( x = 9 \) back into the original equation: \[ 9^{3/2} = (9^{1/2})^3 = 3^3 = 27. \] Since both sides match, our solution is verified as \( x = 9 \).