If $\sqrt{1 + \frac{55}{729}} = 1 + \frac{x}{27}$, then the value of $x$ is

Concept & Formula To solve equations with variables outside a radical, first simplify the numeric expression under the square root, compute its exact root, and then algebraically isolate the unknown variable. Step-by-Step Solution * Given equation: $\sqrt{1 + \frac{55}{729}} = 1 + \frac{x}{27}$ * Simplify the expression under the square root by finding a common denominator: $1 + \frac{55}{729} = \frac{729 + 55}{729} = \frac{784}{729}$ * The equation becomes: $\sqrt{\frac{784}{729}} = 1 + \frac{x}{27}$ * Extract the square root of the numerator and the denominator. We know that $27^2 = 729$. For $784$, since $25^2 = 625$ and $30^2 = 900$, and it ends in $4$, the root is $28$. $\frac{\sqrt{784}}{\sqrt{729}} = \frac{28}{27}$ * Substitute this back into the equation: $\frac{28}{27} = 1 + \frac{x}{27}$ * Subtract $1$ from both sides: $\frac{28}{27} - 1 = \frac{x}{27}$ * $\frac{28 - 27}{27} = \frac{x}{27}$ * $\frac{1}{27} = \frac{x}{27}$ * Since the denominators are equal, the numerators must be equal: $x = 1$ Exam Strategy & Shortcut Recognize the pattern $\sqrt{1 + \frac{k}{d^2}} = \frac{n}{d}$. Here, the denominator is $d = 27$ ($27^2 = 729$). The numerator under the root becomes $729 + 55 = 784$. The root is $\frac{28}{27}$. The right side is structured as $1 + \frac{x}{27} = \frac{27 + x}{27}$. Equating numerators: $28 = 27 + x$, which instantly gives $x = 1$. Common Pitfall Students often try to square both sides of the initial equation right away, resulting in $1 + \frac{55}{729} = (1 + \frac{x}{27})^2$. Expanding the right side creates a quadratic equation $(1 + \frac{2x}{27} + \frac{x^2}{729})$, making the math unnecessarily tedious and increasing the risk of calculation errors. Final Answer Therefore, the correct answer is **1**.