pplify: \( \frac{\left(x^{2}-\frac{1}{y^{2}}\right)^{a} \times\left(x-\frac{1}{y}\right)^{b-a}}{\left(y^{2}-\frac{1}{x^{2}}\right)^{b} \times\left(y+\frac{1}{x}\right)^{a-b}} \)

To simplify the expression \( \frac{\left(x^{2}-\frac{1}{y^{2}}\right)^{a} \times\left(x-\frac{1}{y}\right)^{b-a}}{\left(y^{2}-\frac{1}{x^{2}}\right)^{b} \times\left(y+\frac{1}{x}\right)^{a-b}} \), we can start by rewriting the components using algebraic identities. Recall that \( x^{2} - \frac{1}{y^{2}} = \left(x - \frac{1}{y}\right)\left(x + \frac{1}{y}\right) \) and \( y^{2} - \frac{1}{x^{2}} = \left(y - \frac{1}{x}\right)\left(y + \frac{1}{x}\right) \). Substituting those identities into the expression, we can organize terms: 1. Change \( \left(x^{2} - \frac{1}{y^{2}}\right)^{a} \) into \( \left(\left(x - \frac{1}{y}\right)\left(x + \frac{1}{y}\right)\right)^{a} \). 2. Similarly transform \( \left(y^{2} - \frac{1}{x^{2}}\right)^{b} \). After substituting and expanding, we should notice a pattern and cancel similar terms in the numerator and the denominator. Through systematic simplification, you'll find that common factors can often be grouped and reduced significantly. Ultimately, the expression simplifies down to clean ratios involving the variables \( x \) and \( y \), balancing exponents governed by \( a \) and \( b \). Keep an eye on ensuring correctly matching pairs so that you don’t miss vital cancellations!