The value of $$\sqrt{\frac{(0.03)^2 + (0.21)^2 + (0.065)^2}{(0.003)^2 + (0.021)^2 + (0.0065)^2}}$$ is

### Concept & Formula Express the terms in the denominator as a scaled version of the terms in the numerator by factoring out a common multiple of $10$. ### Step-by-Step Solution Observe the relationship between the numbers in the numerator and the denominator: * $0.003 = \frac{0.03}{10}$ * $0.021 = \frac{0.21}{10}$ * $0.0065 = \frac{0.065}{10}$ Substitute these relationships into the denominator: $$\text{Denominator} = \left(\frac{0.03}{10}\right)^2 + \left(\frac{0.21}{10}\right)^2 + \left(\frac{0.065}{10}\right)^2$$ Factor out $\frac{1}{10^2}$ from the denominator: $$\text{Denominator} = \frac{1}{10^2} \left[ (0.03)^2 + (0.21)^2 + (0.065)^2 \right]$$ Now substitute this back into the original square root expression: $$\sqrt{\frac{(0.03)^2 + (0.21)^2 + (0.065)^2}{\frac{1}{10^2} \left[ (0.03)^2 + (0.21)^2 + (0.065)^2 \right]}}$$ Cancel out the common expression in both the numerator and denominator: $$\sqrt{\frac{1}{\frac{1}{10^2}}} = \sqrt{10^2} = 10$$ ### Exam Strategy & Shortcut Every term in the denominator is exactly $\frac{1}{10}\text{th}$ of its corresponding term in the numerator. Since the terms are squared, the denominator is $\frac{1}{10^2} = \frac{1}{100}\text{th}$ of the numerator. Taking the square root of the reciprocal gives $\sqrt{100} = 10$ instantly. ### Common Pitfall Students frequently choose $10^2$ (or $100$) by forgetting to take the final square root at the end of the simplification process. Always check the outermost operation! ### Final Answer **Therefore, the correct answer is 10.**