The value of $$\sqrt{\frac{0.16}{0.4}}$$ is

### Concept & Formula Simplify the fraction inside the square root first before attempting to extract the root. Eliminate decimals in the numerator and denominator by multiplying by the appropriate power of $10$. ### Step-by-Step Solution Write the expression: $$\sqrt{\frac{0.16}{0.4}}$$ Multiply the numerator and the denominator by $100$ to clear the decimals: $$\sqrt{\frac{16}{40}}$$ Simplify the fraction by dividing both terms by $8$: $$\sqrt{\frac{2}{5}}$$ Convert back to a decimal to evaluate: $$\sqrt{0.4}$$ Evaluate $\sqrt{0.4}$ (as seen in earlier logic): $$\sqrt{0.4} = \sqrt{\frac{40}{100}} = \frac{\sqrt{40}}{10}$$ Since $\sqrt{40} \approx 6.32$, we get: $$\frac{6.32}{10} = 0.632$$ Rounding gives $0.63$. ### Exam Strategy & Shortcut Instantly simplify the internal fraction: $0.16 / 0.4 = 1.6 / 4 = 0.4$. The problem then reduces simply to $\sqrt{0.4}$. Knowing that $0.4 = 40\%$ and $\sqrt{40\%}$ is slightly more than $60\%$ (because $6^2 = 36$), the answer must be around $0.63$. ### Common Pitfall Students sometimes take the square roots of the numerator and denominator separately *before* simplifying: $\sqrt{0.16} = 0.4$, but then they get stuck on $\sqrt{0.4}$ in the denominator, making the calculation unnecessarily complicated. Always simplify the fraction first! ### Final Answer **Therefore, the correct answer is 0.63.**