Questions: Simplify the complex fraction. ((x/y)-(y/x))/((x-y)/(xy)), x ≠ 0, y ≠ 0, x ≠ y

Solution Steps

To simplify the given complex fraction, we need to follow these steps:

1. Simplify the numerator: \(\frac{x}{y} - \frac{y}{x}\).
2. Simplify the denominator: \(\frac{x-y}{xy}\).
3. Divide the simplified numerator by the simplified denominator.

The numerator of the complex fraction is given by: \[ \frac{x}{y} - \frac{y}{x} \] To combine these fractions, we find a common denominator, which is \(xy\): \[ \frac{x^2 - y^2}{xy} \]

The denominator of the complex fraction is: \[ \frac{x - y}{xy} \]

Now, we can express the complex fraction as: \[ \frac{\frac{x^2 - y^2}{xy}}{\frac{x - y}{xy}} \] This simplifies to: \[ \frac{x^2 - y^2}{x - y} \]

The expression \(x^2 - y^2\) can be factored using the difference of squares: \[ x^2 - y^2 = (x - y)(x + y) \] Thus, we have: \[ \frac{(x - y)(x + y)}{x - y} \] Since \(x \neq y\), we can cancel \(x - y\) from the numerator and denominator: \[ x + y \]

Final Answer