Simplfying complex rational expression

Question

I'm trying to simplify
$$ \frac {\dfrac {x}{y} - \dfrac {y}{x}}{y}.$$ My method of trying to solve this is try to simplify the numerator $\frac {x}{y}-\frac{y}{x}$ Then I find the GCD: $xy$, multiply, $\frac{xy}{xy}-\frac{xy}{xy} = 0$, (I know this part is wrong but I don't know why).

Multiply by reciprocal of $y$, $0 \cdot \frac{1}{y} = 0$. Anyway, that's all wrong since the textbook lists the correct answer as $\frac{x^2-y^2}{xy^2}$. How do you simplify this step by step? Thank you.

Answer 1

I know this part is wrong but I don't know why

You start with $\frac{x}{y} - \frac{y}{x}$. You found the correct GCD. It's $xy$. So you want the denominators to be $xy$. Then you need to multiply $\frac{x}{y}$ by $\color{blue}x$ on the top and bottom, and multiply $\frac{y}{x}$ by $\color{red}y$ on the top and bottom:

$$ \frac{x}{y} - \frac{y}{x} = \frac{x}{y} \cdot \color{blue}{\frac{x}{x}} - \frac{y}{x} \cdot \color{red}{\frac{y}{y}} = \frac{x^2}{yx} - \frac{y^2}{xy} = \frac{x^2-y^2}{xy}$$

Then you can continue:

$$\frac{\frac{x}{y} - \frac{y}{x}}{y} = \frac{\frac{x^2-y^2}{xy}}{y} = \frac{x^2-y^2}{xy} \cdot \frac{1}{y} = \frac{x^2-y^2}{xy^2}$$

Here's another way to do it from the beginning:

\begin{align} \frac{\frac{x}{y} - \frac{y}{x}}{y} &= \frac{\frac{x}{y} - \frac{y}{x}}{y} \cdot \frac{xy}{xy}\\[0.3cm] &= \frac{\left(\frac{x}{y} - \frac{y}{x}\right) xy}{y \cdot xy}\\[0.3cm] &= \frac{\frac{x}{y} \cdot xy - \frac{y}{x} \cdot xy}{xy^2}\\[0.3cm] &= \frac{x^2 - y^2}{xy^2} \end{align}

Answer 2

The numerator is $$ \frac{x}{y}-\frac{y}{x}=\frac{x^2}{xy}-\frac{y^2}{xy}=\frac{x^2-y^2}{xy} $$ so you finally get $$ \frac{x^2-y^2}{xy}\,\frac{1}{y}=\frac{x^2-y^2}{xy^2} $$ Note that the reciprocal of $y$ is $\dfrac{1}{y}$.