# Question 1a422

Oct 18, 2017

$\frac{12 {a}^{- 2} b {c}^{- 2}}{3 {a}^{3} {b}^{- 3}} = \textcolor{red}{\frac{4 {b}^{4}}{{a}^{5} {c}^{2}}}$

#### Explanation:

Remember that $\textcolor{b l u e}{{b}^{- a} = \frac{1}{{b}^{a}}}$ and $\textcolor{b l u e}{\frac{1}{b} ^ \left(- a\right) = {b}^{a}}$

(12a^(-2)bc^(-2))/(3a^3b^(-3)# can be decomposed as:
$\textcolor{w h i t e}{\text{XXX}} 12 \cdot {a}^{- 2} \cdot b \cdot {c}^{- 2} \cdot \frac{1}{3} \cdot \frac{1}{{a}^{3}} \cdot \frac{1}{{b}^{- 3}}$

Using our remembrance
$\textcolor{w h i t e}{\text{XXX}} = 12 \cdot \frac{1}{{a}^{2}} \cdot b \cdot \frac{1}{{c}^{2}} \cdot \frac{1}{3} \cdot \frac{1}{{a}^{3}} \cdot {b}^{3}$

Grouping like factors:
$\textcolor{w h i t e}{\text{XXX}} = \underbrace{12 \cdot \frac{1}{3}} \cdot \underbrace{\frac{1}{{a}^{2}} \cdot \frac{1}{{a}^{3}}} \cdot \underbrace{b \cdot {b}^{3}} \cdot \underbrace{\frac{1}{{c}^{2}}}$

Combining like factors:
$\textcolor{w h i t e}{\text{XXX}} = 4 \cdot \frac{1}{{a}^{5}} \cdot {b}^{4} \cdot \frac{1}{{c}^{2}}$

Writing in "compressed form"
$\textcolor{w h i t e}{\text{XXX}} = \frac{4 {b}^{4}}{{a}^{5} {c}^{2}}$