If $\frac{52}{x} = \sqrt{\frac{169}{289}}$, the value of $x$ is

Concept & Formula Resolve the square roots of the known perfect squares on the right side of the equation, then use cross-multiplication to isolate the unknown variable $x$. $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ Step-by-Step Solution * **Given:** $\frac{52}{x} = \sqrt{\frac{169}{289}}$ * **Calculation:** Recognize that $169$ and $289$ are perfect squares. * $\sqrt{169} = 13$ and $\sqrt{289} = 17$. * Substitute these into the equation: $\frac{52}{x} = \frac{13}{17}$ * Isolate $x$ using cross-multiplication: $x = \frac{52 \times 17}{13}$ * Simplify the fraction. Notice that $52$ is a multiple of $13$ ($13 \times 4 = 52$): $x = 4 \times 17$ * $x = 68$ Exam Strategy & Shortcut Memorizing squares up to $30$ is highly recommended. Immediately convert the RHS to $13/17$. Notice the horizontal relationship between the numerators: $13 \times 4 = 52$. Apply the same multiplier to the denominator: $17 \times 4 = 68$. Common Pitfall Failing to recognize $289$ as the perfect square of $17$, leading to unnecessary estimation or getting stuck on the simplification step. Final Answer **Therefore, the correct answer is 68.**