Question

Divide Rational Expressions:$\frac{24b^2}{2b\hspace{0.17em}-\hspace{0.17em}4}÷\frac{12b^2+36b}{b^2\hspace{0.17em}-\hspace{0.17em}11b\hspace{0.17em}+\hspace{0.17em}18}$

Short answer

The required answer is$\frac{b\left(b-9\right)}{b+3}$.

Step-by-step solution

Step 1. Given information 

The given expression is$\frac{24b^2}{2b\hspace{0.17em}-\hspace{0.17em}4}÷\frac{12b^2+36b}{b^2\hspace{0.17em}-\hspace{0.17em}11b\hspace{0.17em}+\hspace{0.17em}18}$.

Step 2.  Simplify the expression. 

$$\begin{gathered} \frac{24b^2}{2b\hspace{0.17em}-\hspace{0.17em}4}÷\frac{12b^2+36b}{b^2\hspace{0.17em}-\hspace{0.17em}11b\hspace{0.17em}+\hspace{0.17em}18} \\ =\frac{\frac{24b^2}{2b-4}}{\frac{12b^2+36b}{b^2-11b+18}} \end{gathered}$$

Apply the fraction rule.

$$\begin{gathered} \frac{24b^2\left(b^2-11b+18\right)}{\left(2b-4\right)\left(12b^2+36b\right)} \\ =\frac{2·\hspace{0.17em}12b^2\left(b^2-11b+18\right)}{2\left(b-2\right)\left(12b^2+36b\right)} \\ =\frac{12b^2\left(b-2\right)\left(b-9\right)}{\left(b-2\right)\left(12b^2+36b\right)} \\ =\frac{12b^2\left(b-9\right)}{12b^2+36b} \\ =\frac{12b^2\left(b-9\right)}{12b\left(b+3\right)} \\ =\frac{b^2\left(b-9\right)}{b\left(b+3\right)} \\ =\frac{b\left(b-9\right)}{b+3} \end{gathered}$$