# How do you multiply and simplify -\frac { 5m ^ { 5} } { 63n } \cdot \frac { 81m n } { 40m ^ { 7} }?

Apr 27, 2018

See a solution process below:

#### Explanation:

First, rewrite the expression as:

$\frac{- 5 \cdot 81}{63 \cdot 40} \left(\frac{{m}^{5} \cdot m}{m} ^ 7\right) \left(\frac{n}{n}\right) \implies$

$\frac{- 5 \cdot 81}{63 \cdot 40} \left(\frac{{m}^{5} \cdot m}{m} ^ 7\right) 1 \implies$

$\frac{- 5 \cdot 81}{63 \cdot 40} \left(\frac{{m}^{5} \cdot m}{m} ^ 7\right) \implies$

$\frac{- 5 \cdot 9 \cdot 9}{9 \cdot 7 \cdot 8 \cdot 5} \left(\frac{{m}^{5} \cdot m}{m} ^ 7\right) \implies$

$\frac{- \textcolor{red}{\cancel{\textcolor{b l a c k}{5}}} \cdot \textcolor{b l u e}{\cancel{\textcolor{b l a c k}{9}}} \cdot 9}{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{9}}} \cdot 7 \cdot 8 \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{5}}}} \left(\frac{{m}^{5} \cdot m}{m} ^ 7\right) \implies$

$- \frac{9}{56} \left(\frac{{m}^{5} \cdot m}{m} ^ 7\right)$

Next, use these rules for exponents to simplify the $m$ terms in the numerator:

$a = {a}^{\textcolor{b l u e}{1}}$ and ${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$

$- \frac{9}{56} \left(\frac{{m}^{5} \cdot m}{m} ^ 7\right) \implies$

$- \frac{9}{56} \left(\frac{{m}^{\textcolor{red}{5}} \cdot {m}^{\textcolor{b l u e}{1}}}{m} ^ 7\right) \implies$

$- \frac{9}{56} \left({m}^{\textcolor{red}{5} + \textcolor{b l u e}{1}} / {m}^{7}\right) \implies$

$- \frac{9}{56} \left({m}^{6} / {m}^{7}\right)$

Now, use these rules of exponents to complete the simplification:

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

$- \frac{9}{56} \left({m}^{\textcolor{red}{6}} / {m}^{\textcolor{b l u e}{7}}\right) \implies$

$- \frac{9}{56} \left(\frac{1}{m} ^ \left(\textcolor{b l u e}{7} - \textcolor{red}{6}\right)\right) \implies$

$- \frac{9}{56} \left(\frac{1}{m} ^ 1\right) \implies$

$- \frac{9}{56} \left(\frac{1}{m}\right) \implies$

$- \frac{9}{56 m}$