Question

In the following exercises, simplify each complex rational expression by writing it as division.

$\frac{\frac{1}{r}-\frac{1}{t}}{\frac{1}{r^2}-\frac{1}{t^2}}$

Short answer

The simplified expression is$\frac{rt}{t+r}$.

Step-by-step solution

Step 1. Given Information.   

We have:

$\frac{\frac{1}{r}-\frac{1}{t}}{\frac{1}{r^2}-\frac{1}{t^2}}$

Step 2. Simplify the numerator:  

The numerator is $\frac{1}{r}-\frac{1}{t}$.

The least common multiplier of $r,t$is $rt.$

Adjust the fractions based on the LCM:

$$\begin{gathered} =\frac{t}{rt}-\frac{r}{rt} \\ =\frac{t-r}{rt} \end{gathered}$$

Step 3. Simplify the denominator.    

The denominator is $\frac{1}{r^2}-\frac{1}{t^2}$.

The least common multiplier of $r^2,t^2$is $r^2{t}^2$.

Adjust the fractions based on the LCM:

$$\begin{gathered} =\frac{t^2}{r^2{t}^2}-\frac{r^2}{r^2{t}^2} \\ =\frac{t^2-r^2}{r^2{t}^2} \end{gathered}$$

Step 4. Substitute the simplified numerator and denominator into the original expression.  

We have:

$\frac{\frac{1}{r}-\frac{1}{t}}{\frac{1}{r^2}-\frac{1}{t^2}}=\frac{\frac{t-r}{rt}}{\frac{t^2-r^2}{r^2{t}^2}}$

Apply fraction rule:$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a·\hspace{0.17em}d}{b·\hspace{0.17em}c}$

$$\begin{gathered} =\frac{\left(t-r\right)r^2{t}^2}{rt\left(t^2-r^2\right)} \\ =\frac{rt}{t+r} \end{gathered}$$