Question

The expression $\frac{(x + \frac{1}{y}) {}^{a} \times (x - \frac{1}{y}) {}^{b} }{(y + \frac{1}{x}) {}^{a} \times (y - \frac{1}{x} ) {}^{b} }$ reduces to ?​

Answer 1

$\frac{(x + \frac{1}{y}) {}^{a} \times (x - \frac{1}{y}) {}^{b} }{(y + \frac{1}{x}) {}^{a} \times (y - \frac{1}{x} ) {}^{b} }$

⠀

$= \frac{( \frac{xy + 1}{y} ) {}^{a} \times ( \frac{xy - 1}{y} ) {}^{b} }{( \frac{xy + 1}{x} ) {}^{a} \times ( \frac{xy - 1}{x} ) {}^{b} }$

$= \frac{ \frac{(x + y) {}^{a} }{y {}^{1} } \times \frac{(xy - 1) {}^{b} }{y {}^{b} } }{ \frac{(xy + 1) {}^{a} }{x {}^{a} } \times \frac{(xy - 1) {}^{b} }{x {}^{b} } }$

$= \frac{ \frac{1}{y {}^{a} } \times \frac{1}{y {}^{b} } }{ \frac{1}{x {}^{a} } \times \frac{1}{x {}^{b} } } \\ \\ = \frac{x {}^{a + b} }{y {}^{a + b} } \\ \\ = ( \frac{x}{y} ) {}^{a + b}$