Question

Divide Rational Expressions:$\frac{27y^2}{3y\hspace{0.17em}-\hspace{0.17em}21}\hspace{0.17em}÷\frac{3y^2+18}{y^2+13y+42}$.

Short answer

The required answer is$\frac{3y^2\left(y^2+13y+42\right)}{\left(y-7\right)\left(y^2+6\right)}$.

Step-by-step solution

Step 1. Given information.

The given expression is$\frac{27y^2}{3y\hspace{0.17em}-\hspace{0.17em}21}\hspace{0.17em}÷\frac{3y^2+18}{y^2+13y+42}$.

Step 2.  Simplify the expression. 

$$\begin{gathered} \frac{27y^2}{3y\hspace{0.17em}-\hspace{0.17em}21}\hspace{0.17em}÷\frac{3y^2+18}{y^2+13y+42} \\ \frac{\frac{27y^2}{3y-21}}{\frac{3y^2+18}{y^2+13y+42}} \end{gathered}$$

Apply the fraction rule.

$$\begin{gathered} =\frac{27y^2\left(y^2+13y+42\right)}{\left(3y-21\right)\left(3y^2+18\right)} \\ =\frac{3y^2\left(y^2+13y+42\right)}{\left(y-7\right)\left(y^2+6\right)} \end{gathered}$$