Question

Simplify a Complex Rational Expression by Using the LCD.

In the following exercises, simplify.

$\frac{\frac{c}{d}+\frac{1}{d}}{\frac{1}{d}-\frac{d}{c}}$

Short answer

The simplified form of the given expression is $\frac{c(c+1)}{c-d^2}$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$\frac{\frac{c}{d}+\frac{1}{d}}{\frac{1}{d}-\frac{d}{c}}$

Step 2. Rewrite the expression and then simplify.

The LCD of all the fractions in the whole expression is $cd$. Clear the fractions by multiplying the numerator and denominator by that LCD.

$$\begin{gathered} =\frac{cd·\left(\frac{c}{d}+\frac{1}{d}\right)}{cd·\left(\frac{1}{d}-\frac{d}{c}\right)} \\ =\frac{cd·\frac{c}{d}+cd·\frac{1}{d}}{cd·\frac{1}{d}-cd·\frac{d}{c}} \\ =\frac{c^2+c}{c-d^2} \\ =\frac{c(c+1)}{c-d^2} \end{gathered}$$

Step 3. Conclusion.

The simplified form of the given expression is $\frac{c(c+1)}{c-d^2}$.