# How do you simplify (12m^-4p^2)/(-15m^3p^-9)?

Jul 18, 2017

See a solution process below:

#### Explanation:

First, rewrite the expression as:

$\left(\frac{12}{-} 15\right) \left({m}^{-} \frac{4}{m} ^ 3\right) \left({p}^{2} / {p}^{-} 9\right) \implies - \frac{3 \times 4}{3 \times 5} \left({m}^{-} \frac{4}{m} ^ 3\right) \left({p}^{2} / {p}^{-} 9\right) \implies$

$- \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \times 4}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \times 5} \left({m}^{-} \frac{4}{m} ^ 3\right) \left({p}^{2} / {p}^{-} 9\right) \implies$

$- \frac{4}{5} \left({m}^{-} \frac{4}{m} ^ 3\right) \left({p}^{2} / {p}^{-} 9\right)$

Next, use this rule for exponents to simplify the $m$ terms:

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

$- \frac{4}{5} \left({m}^{\textcolor{red}{- 4}} / {m}^{\textcolor{b l u e}{3}}\right) \left({p}^{2} / {p}^{-} 9\right) \implies - \frac{4}{5} \left(\frac{1}{m} ^ \left(\textcolor{b l u e}{3} - \textcolor{red}{- 4}\right)\right) \left({p}^{2} / {p}^{-} 9\right) \implies$

$- \frac{4}{5} \left(\frac{1}{m} ^ \left(\textcolor{b l u e}{3} + \textcolor{red}{4}\right)\right) \left({p}^{2} / {p}^{-} 9\right) \implies - \frac{4}{5} \left(\frac{1}{m} ^ 7\right) \left({p}^{2} / {p}^{-} 9\right) \implies$

$- \frac{4}{5 {m}^{7}} \left({p}^{2} / {p}^{-} 9\right)$

Now, use this rule for exponents to simplify the $p$ terms:

${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} - \textcolor{b l u e}{b}}$

$- \frac{4}{5 {m}^{7}} \left({p}^{\textcolor{red}{2}} / {p}^{\textcolor{b l u e}{- 9}}\right) \implies - \frac{4}{5 {m}^{7}} {p}^{\textcolor{red}{2} - \textcolor{b l u e}{- 9}} \implies - \frac{4}{5 {m}^{7}} {p}^{\textcolor{red}{2} + \textcolor{b l u e}{9}} \implies$

$- \frac{4}{5 {m}^{7}} {p}^{11} \implies$

$- \frac{4 {p}^{11}}{5 {m}^{7}}$

Jul 18, 2017

Factor and divide out common terms

#### Explanation:

$12 {m}^{-} 4 {p}^{2} = 3 \times 4 \times \frac{1}{m} ^ 4 \times {p}^{2}$

$15 {m}^{3} {p}^{-} 9 = 3 \times 5 \times {m}^{3} \times \frac{1}{p} ^ 9$

Putting this a complex fraction gives

 { (3 xx 4 xx p^2)/(m^4)}/{(3 xx 5 xx m^3)/(p^9)

Multiplying both the top and the bottom by the inverse of the bottom gives

$\frac{\left(3 \times 4 \times {p}^{2}\right) \times \left({p}^{9}\right)}{\left({m}^{4}\right) \times \left(3 \times 5 \times {m}^{3}\right)}$

dividing out common terms and combining common terms gives

$\frac{4 \times {p}^{11}}{5 \times {m}^{7}}$

Jul 18, 2017

$\frac{4 {p}^{11}}{- 5 {m}^{7}}$
$\frac{12 {m}^{-} 4 {p}^{2}}{- 15 {m}^{3} {p}^{-} 9}$
$\therefore = \frac{{\cancel{12}}^{4} {p}^{2 + 9}}{{\cancel{- 15}}^{-} 5 {m}^{3 + 4}}$
$\therefore = \frac{4 {p}^{11}}{- 5 {m}^{7}}$