Question

Divide Rational Expressions:$\frac{16a^2}{4a\hspace{0.17em}+\hspace{0.17em}36}÷\frac{4a^2-24a}{a^2\hspace{0.17em}+\hspace{0.17em}4a\hspace{0.17em}-\hspace{0.17em}45}$.

Short answer

The required answer is$\frac{a\left(a-5\right)}{a-6}$.

Step-by-step solution

Step 1. Given information. 

The given expression is$\frac{16a^2}{4a\hspace{0.17em}+\hspace{0.17em}36}÷\frac{4a^2-24a}{a^2\hspace{0.17em}+\hspace{0.17em}4a\hspace{0.17em}-\hspace{0.17em}45}$.

Step 2.  Simplify the expression. 

$$\begin{gathered} \frac{16a^2}{4a\hspace{0.17em}+\hspace{0.17em}36}÷\frac{4a^2-24a}{a^2\hspace{0.17em}+\hspace{0.17em}4a\hspace{0.17em}-\hspace{0.17em}45} \\ =\frac{\frac{16a^2}{4a+36}}{\frac{4a^2-24a}{a^2+4a-45}} \end{gathered}$$

Apply the fraction rule.

$$\begin{gathered} =\frac{16a^2\left(a^2+4a-45\right)}{\left(4a+36\right)\left(4a^2-24a\right)} \\ =\frac{4·\hspace{0.17em}4a^2\left(a^2+4a-45\right)}{4\left(a+9\right)\left(4a^2-24a\right)} \\ =\frac{4a^2\left(a-5\right)\left(a+9\right)}{\left(a+9\right)\left(4a^2-24a\right)} \\ =\frac{4a^2\left(a-5\right)}{4a^2-24a} \\ =\frac{4a^2\left(a-5\right)}{4a\left(a-6\right)} \\ =\frac{a^2\left(a-5\right)}{a\left(a-6\right)} \\ =\frac{a\left(a-5\right)}{a-6} \end{gathered}$$