Question

In the following exercises, simplify.

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

Short answer

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

Step-by-step solution

Step 1. Given information.

The given expression is:

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

Step 2. Concept.

The following concept is required for simplification:

$\frac{A}{B}÷\frac{C}{D}=\frac{A}{B}·\frac{D}{C}$

Step 3. Rewrite the expression and then simplify.

Rewrite the complex fraction as division.

$\frac{5}{c^2+5c-14}÷\frac{10}{c+7}$

Rewrite as the product of first times the reciprocal of the second.

$$\begin{gathered} =\frac{5}{c^2+5c-14}·\frac{c+7}{10} \\ =\frac{5}{c^2+7c-2c-14}·\frac{c+7}{10} \\ =\frac{5}{c(c+7)-2(c+7)}·\frac{c+7}{10} \\ =\frac{5(c+7)}{(c+7)(c-2)\left(2\right)\left(5\right)} \\ =\frac{1}{2(c-2)} \end{gathered}$$

Step 4. Conclusion.

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