Question

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

$\frac{\frac{c}{c^2+5c-14}}{\frac{10}{c+7}}$

Short answer

The simplified expression is$\frac{c}{10\left(c-2\right)}$.

Step-by-step solution

Step 1. Given Information.

We have:

$\frac{\frac{c}{c^2+5c-14}}{\frac{10}{c+7}}$

Step 2. Find the LCD of all fractions in complex rational expression.   

The least common divisor (LCD) is $\left(c+7\right)\left(c-2\right)$.

Multiply the numerator and denominator by the LCD:

$=\frac{\left(c+7\right)\left(c-2\right)\frac{c}{c^2+5c-14}}{\left(c+7\right)\left(c-2\right)\frac{10}{c+7}}$

Step 3. Simplify the expression. 

Further, calculate:

$$\begin{gathered} =\frac{\left(c+7\right)\left(c-2\right)\frac{c}{\left(c^2+5c-14\right)}}{\left(c+7\right)\left(c-2\right)\frac{10}{\left(c+7\right)}} \\ =\frac{\left(c+7\right)\left(c-2\right)\frac{c}{\left(c^2+7c-2c-14\right)}}{\left(c+7\right)\left(c-2\right)\frac{10}{\left(c+7\right)}} \\ =\frac{\left(c+7\right)\left(c-2\right)\frac{c}{\left(c\left(c+7\right)-2\left(c+7\right)\right)}}{\left(c+7\right)\left(c-2\right)\frac{10}{\left(c+7\right)}} \\ =\frac{\bcancel{\left(c+7\right)\left(c-2\right)}\frac{c}{\bcancel{\left(c+7\right)\left(c-2\right)}}}{\bcancel{\left(c+7\right)}\left(c-2\right)\frac{10}{\bcancel{\left(c+7\right)}}} \\ =\frac{c}{10\left(c-2\right)} \end{gathered}$$