$\sqrt{\frac{x}{169}} = \frac{54}{39}$

Concept & Formula Separate the numerator and denominator under a radical sign to evaluate known perfect squares immediately. Then use basic cross-multiplication. $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ Step-by-Step Solution * **Given:** $\sqrt{\frac{x}{169}} = \frac{54}{39}$ * **Calculation:** Apply the square root separately to the numerator and denominator: $\frac{\sqrt{x}}{\sqrt{169}} = \frac{54}{39}$ * Substitute the known square root ($\sqrt{169} = 13$): $\frac{\sqrt{x}}{13} = \frac{54}{39}$ * Simplify the right side of the equation. Notice that both $54$ and $39$ are divisible by $3$: $\frac{54}{39} = \frac{18}{13}$ * The equation is now: $\frac{\sqrt{x}}{13} = \frac{18}{13}$ * Since the denominators are identical, the numerators must be equal: $\sqrt{x} = 18$ * Square both sides to find $x$: $x = 18^2$ $x = 324$ Exam Strategy & Shortcut Spotting the relationship between the denominators is key. $\sqrt{169}$ is $13$. Look at the RHS fraction: $54/39$. Recognize that $39$ is exactly $13 \times 3$. So, reducing $54/39$ by a factor of $3$ immediately gives $18/13$. Since the base $13$ matches on both sides, $\sqrt{x} = 18 \implies x = 324$. Common Pitfall Squaring the entire original equation ($\frac{x}{169} = \frac{2916}{1521}$) before simplifying the fractions. This results in incredibly large and cumbersome numbers that invite calculation errors. Final Answer **Therefore, the correct answer is 324.**