$$\sqrt{\frac{9.5 \times .085}{.0017 \times .19}}$$ equals

### Concept & Formula Equalize the number of decimal places in the numerator and the denominator by multiplying by powers of $10$, then simplify the remaining expression under the radical. ### Step-by-Step Solution Count the decimal places: * Numerator: $1 \text{ (from 9.5)} + 3 \text{ (from 0.085)} = 4$ decimal places. * Denominator: $4 \text{ (from 0.0017)} + 2 \text{ (from 0.19)} = 6$ decimal places. To remove the decimals, multiply the numerator by $10^2$ (or $100$) to balance the $6$ decimal places of the denominator: $$\sqrt{\frac{95 \times 85 \times 100}{17 \times 19}}$$ Now simplify the integers: * $\frac{95}{19} = 5$ * $\frac{85}{17} = 5$ Substitute these back into the radical: $$\sqrt{5 \times 5 \times 100} = \sqrt{25 \times 100} = 5 \times 10 = 50$$ ### Exam Strategy & Shortcut Notice that $95/19 = 5$ and $85/17 = 5$. The core numbers give $5 \times 5 = 25$, whose square root is $5$. The choices are all variants of $5$ shifted by decimals. The denominator has $2$ more decimal places than the numerator, which shifts a factor of $100$ to the numerator inside the root, translating to a factor of $10$ outside the root: $5 \times 10 = 50$. ### Common Pitfall Forgetting that the extra decimal places are in the denominator can lead to dividing by $10$ instead of multiplying, resulting in an incorrect answer of $0.5$ or $5$. Always remember: more decimals in the denominator means a larger value when simplified. ### Final Answer **Therefore, the correct answer is 50.**