1. The problem is to solve the equation $$x^2 \sqrt{x} = 3$$ for $x$. 2. Rewrite the square root using fractional exponents: $$\sqrt{x} = x^{1/2}$$. 3. Substitute into the equation: $$x^2 \cdot x^{1/2} = 3$$. 4. Use the property of exponents to combine: $$x^{2 + 1/2} = x^{5/2} = 3$$. 5. To solve for $x$, rewrite as: $$x = 3^{\frac{2}{5}}$$. 6. So the solution is $$x = 3^{\frac{2}{5}}$$, which is the fifth root of $3$ squared. 7. This is the only real solution since $x^{5/2}$ is defined for $x \geq 0$ and here $x > 0$.