$\sqrt{0.0169 \times x} = 1.3$

Concept & Logic Eliminate the square root by squaring both sides of the equation, then divide to isolate the variable. Track the decimal shifts carefully. Step-by-Step Solution * **Given:** $\sqrt{0.0169 \times x} = 1.3$ * **Calculation:** Square both sides of the equation to remove the radical: $0.0169 \times x = (1.3)^2$ * Calculate the square of $1.3$. Since $13^2 = 169$, $(1.3)^2 = 1.69$: $0.0169 \times x = 1.69$ * Isolate $x$ by dividing: $x = \frac{1.69}{0.0169}$ * To clear the decimals, multiply both the numerator and denominator by $10000$ (shift the decimal $4$ places right): $x = \frac{16900}{169}$ * Simplify the fraction: $x = 100$ Exam Strategy & Shortcut Evaluate the known root immediately instead of squaring. $\sqrt{0.0169}$ is $0.13$. So, $0.13 \times \sqrt{x} = 1.3$. By simple observation, $0.13$ must be multiplied by $10$ to become $1.3$. So, $\sqrt{x} = 10$. Squaring both sides gives $x = 100$. This mental path is significantly faster than decimal division. Common Pitfall Dividing incorrectly by getting confused by the number of decimal zeros, leading to answers like $10$ or $1000$ instead of the exact multiple $100$. Final Answer **Therefore, the correct answer is 100.**