Question

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

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

Short answer

The simplified expression is$\frac{2}{3\left(d+3\right)}$.

Step-by-step solution

Step 1. Given Information.

We have:

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

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

The least common divisor (LCD) is $\left(d+6\right)\left(d+3\right)$.

Multiply the numerator and denominator by the LCD:

$=\frac{\left(d+6\right)\left(d+3\right)\left(\frac{8}{d^2+9d+18}\right)}{\left(d+6\right)\left(d+3\right)\left(\frac{12}{d+6}\right)}$

Step 3. Simplify the expression.  

Further, calculate:

$$\begin{gathered} =\frac{\left(d+6\right)\left(d+3\right)\left(\frac{8}{d^2+3d+6d+18}\right)}{\left(d+6\right)\left(d+3\right)\left(\frac{12}{d+6}\right)} \\ =\frac{\left(d+6\right)\left(d+3\right)\left(\frac{8}{d\left(d+3\right)+6\left(d+3\right)}\right)}{\left(d+6\right)\left(d+3\right)\left(\frac{12}{d+6}\right)} \\ =\frac{\bcancel{\left(d+6\right)\left(d+3\right)}\left(\frac{8}{\bcancel{\left(d+3\right)\left(d+6\right)}}\right)}{\bcancel{\left(d+6\right)}\left(d+3\right)\left(\frac{12}{\bcancel{d+6}}\right)} \\ =\frac{8}{12\left(d+3\right)} \\ =\frac{2}{3\left(d+3\right)} \end{gathered}$$