# How do you find the quotient 12/(3x^2)div6/x?

Feb 12, 2017

See the entire solution process below:

#### Explanation:

First, we can rewrite this expression as:

$\frac{\frac{12}{3 {x}^{2}}}{\frac{6}{x}}$

Then, we can use this rule for dividing fractions:

$\frac{\frac{\textcolor{red}{a}}{\textcolor{b l u e}{b}}}{\frac{\textcolor{g r e e n}{c}}{\textcolor{p u r p \le}{d}}} = \frac{\textcolor{red}{a} \times \textcolor{p u r p \le}{d}}{\textcolor{b l u e}{b} \times \textcolor{g r e e n}{c}}$

Substituting the terms from the problem gives:

$\frac{\frac{\textcolor{red}{12}}{\textcolor{b l u e}{3 {x}^{2}}}}{\frac{\textcolor{g r e e n}{6}}{\textcolor{p u r p \le}{x}}} = \frac{\textcolor{red}{12} \times \textcolor{p u r p \le}{x}}{\textcolor{b l u e}{3 {x}^{2}} \times \textcolor{g r e e n}{6}} = \frac{12 x}{18 {x}^{2}}$

Next, let's simplify the constants:

$\frac{12 x}{18 {x}^{2}} = \frac{\left(6 \times 2\right) x}{\left(6 \times 3\right) {x}^{2}} = \frac{\left(\textcolor{red}{\cancel{\textcolor{b l a c k}{6}}} \times 2\right) x}{\left(\textcolor{red}{\cancel{\textcolor{b l a c k}{6}}} \times 3\right) {x}^{2}} = \frac{2 x}{3 {x}^{2}}$

Now, we can use these rules for exponents to complete the solution:

${a}^{\textcolor{red}{1}} = a$ or $a = {a}^{\textcolor{red}{1}}$ and ${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = \frac{1}{x} ^ \left(\textcolor{b l u e}{b} - \textcolor{red}{a}\right)$

$\frac{2 x}{3 {x}^{2}} = \frac{2 {x}^{\textcolor{red}{1}}}{3 {x}^{\textcolor{b l u e}{2}}} = \frac{2}{3 {x}^{\textcolor{b l u e}{2} - \textcolor{red}{1}}} = \frac{2}{3 {x}^{1}} = \frac{2}{3 x}$