Question

Divide Rational Expressions:$\frac{3s^2}{s^2\hspace{0.17em}-\hspace{0.17em}16}÷\frac{s^3\hspace{0.17em}+\hspace{0.17em}4s^2\hspace{0.17em}+\hspace{0.17em}16s}{s^3\hspace{0.17em}-\hspace{0.17em}64}$.

Short answer

The required answer is$\frac{3s}{s+4}$.

Step-by-step solution

Step 1. Given information.

The given expression is$\hspace{0.17em}\hspace{0.17em}\frac{3s^2}{s^2\hspace{0.17em}-\hspace{0.17em}16}÷\frac{s^3\hspace{0.17em}+\hspace{0.17em}4s^2\hspace{0.17em}+\hspace{0.17em}16s}{s^3\hspace{0.17em}-\hspace{0.17em}64}$.

Step 2.  Simplify the expression. 

$$\begin{gathered} \hspace{0.17em}\hspace{0.17em}\frac{3s^2}{s^2\hspace{0.17em}-\hspace{0.17em}16}÷\frac{s^3\hspace{0.17em}+\hspace{0.17em}4s^2\hspace{0.17em}+\hspace{0.17em}16s}{s^3\hspace{0.17em}-\hspace{0.17em}64} \\ =\frac{\frac{3s^2}{s^2-16}}{\frac{s^3+4s^2+16s}{s^3-64}} \end{gathered}$$

Apply the fraction rule.

$$\begin{gathered} =\frac{3s^2\left(s^3-64\right)}{\left(s^2-16\right)\left(s^3+4s^2+16s\right)} \\ =\frac{3ss\left(s^3-64\right)}{\left(s^2-16\right)s\left(s^2+4s+16\right)} \\ =\frac{3s\left(s-4\right)\left(s^2+4s+16\right)}{\left(s^2-16\right)\left(s^2+4s+16\right)} \\ =\frac{3s\left(s-4\right)}{s^2-16} \\ =\frac{3s\left(s-4\right)}{\left(s+4\right)\left(s-4\right)} \\ =\frac{3s}{s+4} \end{gathered}$$