If $\sqrt{1369} + \sqrt{.0615 + x} = 37.25$, then $x$ is equal to

Concept & Formula This problem tests the ability to find square roots of large numbers and solve radical equations. Isolate the unknown radical term by subtracting known values, then square both sides. Step-by-Step Solution * Find the square root of $1369$. Since $30^2 = 900$ and $40^2 = 1600$, and the unit digit is $9$, the root must end in $3$ or $7$. $37 \times 37 = 1369$. * Substitute this back into the equation: $37 + \sqrt{0.0615 + x} = 37.25$ * Subtract $37$ from both sides: $\sqrt{0.0615 + x} = 0.25$ * Square both sides to eliminate the square root: $0.0615 + x = (0.25)^2$ * Calculate $(0.25)^2 = 0.0625$ * Solve for $x$: $x = 0.0625 - 0.0615 = 0.001$ * Convert to exponential form: $x = 10^{-3}$ Exam Strategy & Shortcut Memorizing squares up to $40$ or using the unit-digit approximation method instantly yields $\sqrt{1369} = 37$. Once you have $0.25$ on the right side, recognize that $0.25$ is $\frac{1}{4}$, so its square is $\frac{1}{16}$ or $0.0625$, leading straight to $0.001$. Common Pitfall A frequent mistake is misplacing the decimal point when squaring $0.25$. Some might write $0.025$ instead of $0.0625$, leading to an incorrect subtraction and wrong power of 10. Final Answer Therefore, the correct answer is $10^{-3}$.