Question

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

$\frac{\frac{2}{v}+\frac{2}{w}}{\frac{1}{v^2}-\frac{1}{w^2}}$

Short answer

The simplified expression is$\frac{2vw}{w-v}$.

Step-by-step solution

Step 1. Given Information.

We have:

$\frac{\frac{2}{v}+\frac{2}{w}}{\frac{1}{v^2}-\frac{1}{w^2}}$

Step 2. Simplify the numerator:   

The numerator is $\frac{2}{v}+\frac{2}{w}$.

The least common multiplier of $v,w$is $vw.$

Adjust the fractions based on the LCM:

$$\begin{gathered} =\frac{2w}{vw}+\frac{2v}{vw} \\ =\frac{2w+2v}{vw} \end{gathered}$$

Step 3. Simplify the denominator. 

The denominator is $\frac{1}{v^2}-\frac{1}{w^2}$.

The least common multiplier of $v^2,w^2$is $v^2{w}^2$.

Adjust the fractions based on the LCM:

$$\begin{gathered} =\frac{w^2}{v^2{w}^2}-\frac{v^2}{v^2{w}^2} \\ =\frac{w^2-v^2}{v^2{w}^2} \end{gathered}$$

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

We have:

$\frac{\frac{2}{v}+\frac{2}{w}}{\frac{1}{v^2}-\frac{1}{w^2}}=\frac{\frac{2w+2v}{vw}}{\frac{w^2-v^2}{v^2{w}^2}}$

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

$$\begin{gathered} =\frac{\left(2w+2v\right)v^2{w}^2}{vw\left(w^2-v^2\right)} \\ =\frac{2vw}{w-v} \end{gathered}$$