Question

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

$\frac{\frac{2a}{a+4}}{\frac{4a^2}{a^2-16}}$

Short answer

The simplified expression is $\frac{a-4}{2a}$.

Step-by-step solution

Step 1. Given Information.

We have:

$\frac{\frac{2a}{a+4}}{\frac{4a^2}{a^2-16}}$

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

The least common divisor (LCD) is $\left(a-4\right)\left(a+4\right).$

Multiply the numerator and denominator by the LCD:

$=\frac{\left(a-4\right)(a+4)\left(\frac{2a}{a+4}\right)}{\left(a-4\right)(a+4)\left(\frac{4a^2}{a^2-16}\right)}$

Step 3. Simplify the expression.

Further, calculate:

$$\begin{gathered} =\frac{\left(a-4\right)\bcancel{(a+4)}\left(\frac{2a}{\bcancel{\left(a+4\right)}}\right)}{\bcancel{\left(a-4\right)(a+4)}\left(\frac{4a^2}{\bcancel{\left(a-4\right)\left(a+4\right)}}\right)} \\ =\frac{(a-4)2a}{4a^2} \\ =\frac{a-4}{2a} \end{gathered}$$