If $\sqrt{2} = 1.414$, the square root of $\frac{\sqrt{2} - 1}{\sqrt{2} + 1}$ is nearest to

### Concept & Formula The key insight is to rationalize the denominator of the fraction under the square root before attempting to substitute any decimal values. Rationalization involves multiplying the numerator and denominator by the conjugate of the denominator: $$ \frac{a - b}{a + b} \times \frac{a - b}{a - b} = \frac{(a - b)^2}{a^2 - b^2} $$ ### Step-by-Step Solution Given the expression inside the square root: $E = \frac{\sqrt{2} - 1}{\sqrt{2} + 1}$ Rationalize the denominator by multiplying the numerator and denominator by $(\sqrt{2} - 1)$: $E = \frac{(\sqrt{2} - 1)(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)}$ Simplify the numerator as a perfect square and the denominator using $(a+b)(a-b) = a^2 - b^2$: $E = \frac{(\sqrt{2} - 1)^2}{(\sqrt{2})^2 - (1)^2}$ $E = \frac{(\sqrt{2} - 1)^2}{2 - 1} = (\sqrt{2} - 1)^2$ The question asks for the square root of this entire expression: $\sqrt{E} = \sqrt{(\sqrt{2} - 1)^2} = \sqrt{2} - 1$ Now, substitute the given value $\sqrt{2} = 1.414$: $\sqrt{E} = 1.414 - 1 = 0.414$ ### Exam Strategy & Shortcut Whenever you see a fraction of the form $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}}$ inside a square root, immediately know that the denominator will become $a - b$. Since $a - b = 2 - 1 = 1$, the denominator vanishes. The numerator simply becomes the square of the conjugate, which cancels perfectly with the outer square root. You can jump straight to $\sqrt{2} - 1 = 0.414$ mentally in just a few seconds. ### Common Pitfall The most common mistake is forgetting that the question asks for the **square root** of the fraction, not just the value of the fraction itself. Students often rationalize it, get $(\sqrt{2} - 1)^2 \approx 0.171$, and incorrectly guess an answer. Always reread the final prompt requirement. ### Final Answer **Therefore, the correct answer is 0.414.**