1. Stating the problem: Solve the equation $10 = \sqrt{x} 2x^{2}$ for $x$. 2. Rewrite the equation clearly: $10 = 2x^{2} \sqrt{x}$. 3. Express $\sqrt{x}$ as $x^{\frac{1}{2}}$: $10 = 2 x^{2} x^{\frac{1}{2}}$. 4. Combine the powers of $x$: $x^{2} x^{\frac{1}{2}} = x^{2 + \frac{1}{2}} = x^{\frac{5}{2}}$. 5. The equation becomes $10 = 2 x^{\frac{5}{2}}$. 6. Divide both sides by 2: $\frac{10}{2} = x^{\frac{5}{2}}$ which simplifies to $5 = x^{\frac{5}{2}}$. 7. To solve for $x$, raise both sides to the power $\frac{2}{5}$: $5^{\frac{2}{5}} = x$. 8. The exact solution is $x = 5^{\frac{2}{5}}$. 9. Numeric approximation (optional): $5^{\frac{2}{5}} \approx 2.297$. Final answer: $x = 5^{\frac{2}{5}}$.