How do you simplify \frac { x ^ { 2} + x - 12} { x y ^ { 2} } \cdot \frac { x ^ { 7} y } { x ^ { 2} + 7x + 12}?

Mar 26, 2017

$= \setminus \frac{{x}^{6} \left(x - 3\right)}{y \left(x + 3\right)}$

Explanation:

To simplify, we will have to factor the numerator of the first fraction and the denominator of the second fraction.

$\setminus \frac{{x}^{2} + x - 12}{\textcolor{b l u e}{x} \textcolor{g r e e n}{{y}^{2}}} \setminus \cdot \setminus \frac{\textcolor{b l u e}{{x}^{7}} \textcolor{g r e e n}{y}}{{x}^{2} + 7 x + 12}$

$= \setminus \frac{\cancel{\left(x + 4\right)} \left(x - 3\right)}{\textcolor{b l u e}{x} \textcolor{g r e e n}{{y}^{2}}} \setminus \cdot \setminus \frac{\textcolor{b l u e}{{x}^{7}} \textcolor{g r e e n}{y}}{\cancel{\left(x + 4\right)} \left(x + 3\right)}$

We see that the $\left(x + 4\right)$ from the top and the bottom cancel, but also an $x y$ factor (from the denominator of the first fraction and the numerator of the second fraction).

Thus, this simplifies to:

$= \setminus \frac{\left(x - 3\right)}{y} \setminus \cdot \setminus \frac{{x}^{6}}{\left(x + 3\right)}$

$= \setminus \frac{{x}^{6} \left(x - 3\right)}{y \left(x + 3\right)}$