The expression $$ \frac{\left(x + \frac{1}{y}\right)^a \cdot \left(x - \frac{1}{y}\right)^b}{\left(y + \frac{1}{x}\right)^a \cdot \left(y - \frac{1}{x}\right)^b} $$ reduces to

### Concept & Formula To simplify complex fractional expressions, the key insight is to convert the mixed terms into common fractions. This reveals common binomial terms in the numerator and denominator that can be canceled out. $$ x + \frac{1}{y} = \frac{xy + 1}{y} $$ ### Step-by-Step Solution * **Step 1:** Simplify the terms in the numerator by taking a common denominator. $$ \left(x + \frac{1}{y}\right)^a \cdot \left(x - \frac{1}{y}\right)^b = \left(\frac{xy + 1}{y}\right)^a \cdot \left(\frac{xy - 1}{y}\right)^b $$ $$ = \frac{(xy + 1)^a \cdot (xy - 1)^b}{y^{a+b}} $$ * **Step 2:** Similarly, simplify the terms in the denominator. $$ \left(y + \frac{1}{x}\right)^a \cdot \left(y - \frac{1}{x}\right)^b = \left(\frac{xy + 1}{x}\right)^a \cdot \left(\frac{xy - 1}{x}\right)^b $$ $$ = \frac{(xy + 1)^a \cdot (xy - 1)^b}{x^{a+b}} $$ * **Step 3:** Divide the simplified numerator by the simplified denominator. Notice that the large binomial term $ (xy + 1)^a \cdot (xy - 1)^b $ appears in both and will cancel out perfectly. $$ \frac{ \frac{(xy + 1)^a \cdot (xy - 1)^b}{y^{a+b}} }{ \frac{(xy + 1)^a \cdot (xy - 1)^b}{x^{a+b}} } = \frac{x^{a+b}}{y^{a+b}} $$ * **Step 4:** Combine the bases under the common exponent. $$ \frac{x^{a+b}}{y^{a+b}} = \left(\frac{x}{y}\right)^{a+b} $$ ### Exam Strategy & Shortcut **Value Substitution Method:** Pick simple values that don't result in zero denominators. Let $x=2$, $y=1$, $a=1$, and $b=1$. Numerator: $ (2 + 1)^1 \cdot (2 - 1)^1 = 3 \cdot 1 = 3 $ Denominator: $ (1 + \frac{1}{2})^1 \cdot (1 - \frac{1}{2})^1 = \frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4} $ Expression value = $ \frac{3}{3/4} = 4 $ Now check the options with our assumed values: (a) $ (2/1)^{0} = 1 $ (Incorrect) (c) $ (2/1)^{2} = 4 $ (Matches!) This method takes only seconds and safely bypasses all algebra. ### Common Pitfall Students often mix up the positioning of the bases when flipping the fraction in the final step. Since $ y^{a+b} $ is the denominator of the top fraction and $ x^{a+b} $ is the denominator of the bottom fraction, dividing them means $ x $ moves to the top. Always double-check your fraction inversion. ### Final Answer Therefore, the correct answer is **$\left(\frac{x}{y}\right)^{a+b}$**.