Question

which expression is equivalent to the following complex fraction?\\(\frac{\frac{1}{x}-\frac{1}{y}}{\frac{1}{x}+\frac{1}{y}}\\)\\(\frac{y + x}{y - x}\\)\\(\frac{(y - x)(y + x)}{x^2y^2}\\)\\(\frac{x^2y^2}{(y + x)(y + x)}\\)\\(\frac{y - x}{y + x}\\)

Explanation:

Step1: Simplify numerator

Find common denominator $xy$:
$\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy}$

Step2: Simplify denominator

Find common denominator $xy$:
$\frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy}$

Step3: Rewrite complex fraction

Express as division of two fractions:
$\frac{\frac{y - x}{xy}}{\frac{y + x}{xy}}$

Step4: Divide the fractions

Multiply by reciprocal of denominator:
$\frac{y - x}{xy} \times \frac{xy}{y + x} = \frac{y - x}{y + x}$

Answer:

$\frac{y-x}{y+x}$ (the fourth option)