If $\sqrt{.04 \times .4 \times a} = .004 \times .4 \times \sqrt{b}$, then $\frac{a}{b}$ is

Concept & Formula To find the ratio of variables trapped under and outside radicals, isolate the variables by squaring both sides of the equation. Converting small decimals to scientific notation significantly reduces arithmetic errors. Step-by-Step Solution * Given: $\sqrt{0.04 \times 0.4 \times a} = 0.004 \times 0.4 \times \sqrt{b}$ * Square both sides to remove the square roots: $0.04 \times 0.4 \times a = (0.004 \times 0.4)^2 \times b$ * Convert the decimals to powers of 10 for structured calculation: Left Side: $(4 \times 10^{-2}) \times (4 \times 10^{-1}) \times a = 16 \times 10^{-3} \times a$ * Right Side: $((4 \times 10^{-3}) \times (4 \times 10^{-1}))^2 \times b = (16 \times 10^{-4})^2 \times b = 256 \times 10^{-8} \times b$ * Equate and rearrange to find the ratio $\frac{a}{b}$: $16 \times 10^{-3} \times a = 256 \times 10^{-8} \times b$ * $\frac{a}{b} = \frac{256 \times 10^{-8}}{16 \times 10^{-3}}$ * Simplify the fraction and exponents: $\frac{256}{16} \times 10^{-8 - (-3)} = 16 \times 10^{-5}$ Exam Strategy & Shortcut Use decimal digit counting. The left side has a decimal with 3 total decimal places under a root. Squaring the right side (which has $3 + 1 = 4$ decimal places) results in 8 decimal places. When dividing the right side's factor by the left side's factor to find $\frac{a}{b}$, you are essentially doing $10^{-8} \div 10^{-3} = 10^{-5}$. The integer coefficient is $16^2 \div 16 = 16$. Thus, the answer is instantly $16 \times 10^{-5}$. Common Pitfall Attempting to multiply the decimals directly (e.g., $0.004 \times 0.4 = 0.0016$) and then squaring them conventionally ($0.00000256$) usually leads to miscounting the zeros. Always use scientific notation for operations on very small decimals. Final Answer Therefore, the correct answer is $16 \times 10^{-5}$.