Question

The value of $\frac{{(x+\frac{1}{y})}^a{(x-\frac{1}{y})}^b}{{(y+\frac{1}{x})}^a{(y-\frac{1}{x})}^b}$ dis equal to

• A. ${\left(\frac{x}{y}\right)}^{a+b}$
• B. ${\left(\frac{y}{x}\right)}^{a+b}$
• C. ${\left(xy\right)}^{a+b}$
• D. None of these

Answer 1

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

Consider given the expression,

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

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

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

$\Rightarrow \frac{\frac{1}{y^a}\times \frac{1}{y^b}}{\frac{1}{x^a}\times \frac{1}{x^b}}$

$\Rightarrow \frac{x^{a+b}}{y^{a+b}}$

$\Rightarrow {\left(\frac{x}{y}\right)}^{a+b}$

Hence, this is the answer.