1. **State the problem:** Solve the equation $$100\sqrt{x} = \frac{1}{10}(-2)$$. 2. **Rewrite the equation:** The right side simplifies to $$\frac{1}{10} \times (-2) = -\frac{2}{10} = -\frac{1}{5}$$. So the equation becomes: $$100\sqrt{x} = -\frac{1}{5}$$. 3. **Isolate the square root:** Divide both sides by 100: $$\sqrt{x} = \frac{-\frac{1}{5}}{100} = -\frac{1}{5} \times \frac{1}{100} = -\frac{1}{500}$$. Intermediate step with cancellation: $$\sqrt{x} = \frac{\cancel{-1}}{\cancel{500}}$$ (no common factors to cancel here, just showing division). 4. **Analyze the result:** The square root of a real number $x$ is always non-negative, but here we have $$\sqrt{x} = -\frac{1}{500}$$ which is negative. 5. **Conclusion:** There is no real solution because the square root cannot be negative. **Final answer:** No real solution.