Question

In the following exercises, simplify.

$\frac{\frac{8}{d^2+9d+18}}{\frac{12}{d+6}}$

Short answer

The simplified form of the given expression is $\frac{2}{3(d+3)}$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$\frac{\frac{8}{d^2+9d+18}}{\frac{12}{d+6}}$

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.

$\frac{8}{d^2+9d+18}÷\frac{12}{d+6}$

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

role="math" localid="1647763402276" $$\begin{gathered} =\frac{8}{d^2+9d+18}·\frac{d+6}{12} \\ =\frac{8}{d^2+6d+3d+18}·\frac{d+6}{12} \\ =\frac{8}{d(d+6)+3(d+6)}·\frac{d+6}{12} \\ =\frac{4·2·(d+6)}{(d+6)(d+3)(4·3)} \\ =\frac{2}{3(d+3)} \end{gathered}$$

Step 4. Conclusion.

The simplified form of the given expression is $\frac{2}{3(d+3)}$.