Question

In the following exercises, simplify:

$\frac{{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}}{{\displaystyle \frac{1}{p^2}}-{\displaystyle \frac{1}{q^2}}}$.

Short answer

The simplified rational expression is,$\frac{pq}{q-p}$.

Step-by-step solution

Step 1. Given Information.

The rational expression is,

$\frac{{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}}{{\displaystyle \frac{1}{p^2}}-{\displaystyle \frac{1}{q^2}}}$.

Step 2. Simplifying the rational expression.

$\frac{{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}}{{\displaystyle \frac{1}{p^2}}-{\displaystyle \frac{1}{q^2}}}$

The LCD of the expression is $p^2{q}^2$,

$$\begin{gathered} =\frac{p^2{q}^2({\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}})}{p^2{q}^2({\displaystyle \frac{1}{p^2}}-{\displaystyle \frac{1}{q^2}})} \\ =\frac{p^2{q}^2\times {\displaystyle \frac{1}{p}}+p^2{q}^2\times {\displaystyle \frac{1}{q}}}{p^2{q}^2\times {\displaystyle \frac{1}{p^2}}-p^2{q}^2\times {\displaystyle \frac{1}{q^2}}} \\ =\frac{p{q}^2+p^{2}q}{q^2-p^2} \end{gathered}$$

Using $(a^2-b^2)=(a+b)(a-b)$in the denominator,

role="math" localid="1647791789739" $=\frac{pq(q+p)}{(q+p)(q-p)}$

Removing common factors,

role="math" localid="1647791863927" $=\frac{pq}{q-p}$.