If $\sqrt{1 + \frac{x}{169}} = \frac{14}{13}$, then $x$ is equal to

Concept & Formula This involves simplifying algebraic expressions under a square root. Square both sides of the equation to remove the radical, then solve for the unknown variable. Step-by-Step Solution * Given: $\sqrt{1 + \frac{x}{169}} = \frac{14}{13}$ * Square both sides to eliminate the square root: $1 + \frac{x}{169} = \left(\frac{14}{13}\right)^2$ * Compute the squares on the right side: $1 + \frac{x}{169} = \frac{196}{169}$ * Subtract $1$ from both sides: $\frac{x}{169} = \frac{196}{169} - 1$ * Simplify the right side: $\frac{x}{169} = \frac{196 - 169}{169}$ * $\frac{x}{169} = \frac{27}{169}$ * Multiply both sides by $169$: $x = 27$ Exam Strategy & Shortcut Notice the structure $\sqrt{1 + \frac{x}{d^2}} = \frac{n}{d}$. If the difference between $n$ and $d$ is $1$ (like $14$ and $13$), then $x = n^2 - d^2$. More simply, $x = (n-d)(n+d) = (1)(14+13) = 27$. This shortcut directly provides the answer in seconds without manual fraction subtraction. Common Pitfall Students sometimes forget to square the denominator on the right side, calculating it as $\frac{196}{13}$, which completely derails the subsequent fraction subtraction and yields an incorrect $x$. Final Answer Therefore, the correct answer is 27.