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

• A. ${\left(\frac{y}{x}\right)}^{a+b}$
• B. ${\left(\frac{x}{y}\right)}^{a+b}$
• C. ${\left(\frac{y}{x}\right)}^{a-b}$
• D. ${\left(\frac{x}{y}\right)}^{a-b}$

Answer 1

The correct option is B ${\left(\frac{x}{y}\right)}^{a+b}$

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