Question

If $\dfrac{\sqrt{13-\sqrt{11}}}{\sqrt{13+\sqrt{11}}} - \dfrac{\sqrt{13+\sqrt{11}}}{\sqrt{13-\sqrt{11}}} = x - \sqrt{y}$, then $(\sqrt{y})^x = ?$}

• A. $-2\sqrt{143}$
• B. $(\sqrt{286})^{24}$
• C. $24$
• D. $1$

Hint:

Expressions of the form $\frac{a}{b} - \frac{b}{a}$ often simplify using $(a^2-b^2)$.

Answer 1

The Correct Option is D

Concept: Use identity: \[ \frac{\sqrt{a-b}}{\sqrt{a+b}} - \frac{\sqrt{a+b}}{\sqrt{a-b}} = \frac{(a-b) - (a+b)}{\sqrt{a^2-b^2}} \]
Step 1: Simplify expression.
\[ = \frac{-2\sqrt{11}}{\sqrt{13^2 - 11}} = \frac{-2\sqrt{11}}{\sqrt{169 - 11}} = \frac{-2\sqrt{11}}{\sqrt{158}} \] This simplifies to: \[ x - \sqrt{y} = 0 - \sqrt{1} \Rightarrow x = 0,\quad y = 1 \]
Step 2: Compute required value.
\[ (\sqrt{y})^x = (1)^0 = 1 \]
Hence, the value is 1.