Question

Divide Rational Expressions:$\frac{r^2\hspace{0.17em}-\hspace{0.17em}9}{15}÷\frac{r^3\hspace{0.17em}-\hspace{0.17em}27}{5r^2\hspace{0.17em}+\hspace{0.17em}15r\hspace{0.17em}+\hspace{0.17em}45}$

Short answer

The required answer is$\frac{r}{3}+1$.

Step-by-step solution

Step 1. Given information. 

The given expression is$\frac{r^2\hspace{0.17em}-\hspace{0.17em}9}{15}÷\frac{r^3\hspace{0.17em}-\hspace{0.17em}27}{5r^2\hspace{0.17em}+\hspace{0.17em}15r\hspace{0.17em}+\hspace{0.17em}45}$

Step 2.  Simplify the expression. 

$$\begin{gathered} \frac{r^2\hspace{0.17em}-\hspace{0.17em}9}{15}÷\frac{r^3\hspace{0.17em}-\hspace{0.17em}27}{5r^2\hspace{0.17em}+\hspace{0.17em}15r\hspace{0.17em}+\hspace{0.17em}45} \\ =\frac{\frac{r^2-9}{15}}{\frac{r^3-27}{5r^2+15r+45}} \end{gathered}$$

Apply the fraction rule.

$$\begin{gathered} \frac{\left(r^2-9\right)\left(5r^2+15r+45\right)}{15\left(r^3-27\right)} \\ =\frac{\left(r^2-9\right)·\hspace{0.17em}5\left(r^2+3r+9\right)}{5·\hspace{0.17em}3\left(r^3-27\right)} \\ =\frac{\left(r^2-9\right)\left(r^2+3r+9\right)}{3\left(r^3-27\right)} \end{gathered}$$

Step 3. Factor the numerators and denominators and then multiply. 

$$\begin{gathered} =\frac{\left(r^2-9\right)\left(r^2+3r+9\right)}{3\left(r-3\right)\left(r^2+3r+9\right)} \\ =\frac{r^2-9}{3\left(r-3\right)} \\ =\frac{\left(r+3\right)\left(r-3\right)}{3\left(r-3\right)} \\ =\frac{r+3}{3} \\ =\frac{r}{3}+1 \end{gathered}$$