Question

In the following exercises, divide the rational expressions.

$\frac{3s^2}{s^2-16}÷\frac{s^3-4s^2+16s}{s^3-64}$

Short answer

The simplified expression is$\frac{3s^3}{{\left(s-4\right)}^2\left(s+4\right)}.$

Step-by-step solution

Step 1. Given Information

$\frac{3s^2}{s^2-16}÷\frac{s^3-4s^2+16s}{s^3-64}$

$\frac{3s^2}{s^2-16}\times \frac{s^3-4s^2+16s}{s^3-64}$

Step 2. Factors the numerators and denominators

$$\begin{gathered} \frac{3s^2}{\left(s-4\right)\left(s+4\right)}\times \frac{s\left(s^2-4s+16\right)}{s^3-4^3} \\ =\frac{3s^2}{\left(s-4\right)\left(s+4\right)}\times \frac{s\left(s^2-4s+16\right)}{\left(s-4\right)\left(s^2-4s+16\right)} \end{gathered}$$

Step 3. Multiply numerators and denominators and remove common factors

$$\begin{gathered} =\frac{3s\times s}{\left(s-4\right)\left(s+4\right)}\times \frac{s\left(s^2-4s+16\right)}{\left(s-4\right)\left(s^2-4s+16\right)} \\ =\frac{3s^3}{{\left(s-4\right)}^2\left(s+4\right)} \end{gathered}$$