1. We are given the equation $x^2 \sqrt{x} = 3$. 2. Recall that $\sqrt{x} = x^{1/2}$. 3. Rewrite the equation using exponents: $x^2 \cdot x^{1/2} = 3$. 4. Use the property of exponents: $x^a \cdot x^b = x^{a+b}$, so this becomes $x^{2 + 1/2} = 3$. 5. Simplify the exponent: $2 + 1/2 = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$. 6. The equation is now $x^{5/2} = 3$. 7. To solve for $x$, raise both sides to the power of the reciprocal $\frac{2}{5}$: $\left(x^{5/2}\right)^{2/5} = 3^{2/5}$. 8. Simplify the left side: $x^{(5/2)\cdot(2/5)} = x^1 = x$. 9. Therefore, $x = 3^{2/5}$. 10. This is the final answer: $x = 3^{2/5}$.