$\sqrt{\frac{0.361}{0.00169}} = x$

Concept & Math When evaluating the square root of a complex decimal fraction, the safest and most reliable method is to shift the decimal points equally in the numerator and denominator to convert them both into whole numbers before extracting the root. Step-by-Step Solution * Given expression: $\sqrt{\frac{0.361}{0.00169}}$ * Count the decimal places: The numerator has 3 decimal places, and the denominator has 5. * To eliminate decimals completely, multiply both the numerator and denominator by $10^5$ (which is $100,000$). * $\sqrt{\frac{0.361 \times 100000}{0.00169 \times 100000}} = \sqrt{\frac{36100}{169}}$ * Separate the numerator into recognizable perfect squares: $\sqrt{\frac{361 \times 100}{169}}$ * Take the square root of each distinct component: $\sqrt{361} = 19$, $\sqrt{100} = 10$, and $\sqrt{169} = 13$. * Multiply the numerator components: $\frac{19 \times 10}{13} = \frac{190}{13}$ Exam Strategy & Shortcut A highly efficient visual shortcut is to balance the decimal places by padding with zeros. Write $0.361$ as $0.36100$ so it matches the five decimal places of $0.00169$. This visually allows you to drop the decimal points entirely, leaving $\frac{36100}{169}$. The roots of these whole numbers are instantly recognizable as $\frac{190}{13}$. Common Pitfall Students often mismatch the decimal shift. They see the numbers $361$ and $169$ and jump straight to $\frac{19}{13}$ (Option b) without properly accounting for the two extra decimal places in the denominator, which necessitates a compensating factor of $10$ in the numerator. Final Answer Therefore, the correct answer is $\frac{190}{13}$.