# How do you express (-16a^ -3 b^ -5)/(2a^ -4 b^2) with positive exponents?

Jun 23, 2018

See a solution process below:

#### Explanation:

First, rewrite the expression as:

$\frac{- 16}{2} \left({a}^{-} \frac{3}{a} ^ - 4\right) \left({b}^{-} \frac{5}{b} ^ 2\right) \implies - 8 \left({a}^{-} \frac{3}{a} ^ - 4\right) \left({b}^{-} \frac{5}{b} ^ 2\right)$

Next, use these rules for exponents to simplify the $a$ term:

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

$- 8 \left({a}^{\textcolor{red}{- 3}} / {a}^{\textcolor{b l u e}{- 4}}\right) \left({b}^{-} \frac{5}{b} ^ 2\right) \implies$

$- 8 {a}^{\textcolor{red}{- 3} - \textcolor{b l u e}{- 4}} \left({b}^{-} \frac{5}{b} ^ 2\right) \implies$

$- 8 {a}^{\textcolor{red}{- 3} + \textcolor{b l u e}{4}} \left({b}^{-} \frac{5}{b} ^ 2\right) \implies$

$- 8 {a}^{\textcolor{red}{1}} \left({b}^{-} \frac{5}{b} ^ 2\right) \implies$

$- 8 a \left({b}^{-} \frac{5}{b} ^ 2\right)$

Now, use this rule for exponents to simplify the $b$ term:

$- 8 a \left({b}^{\textcolor{red}{- 5}} / {b}^{\textcolor{b l u e}{2}}\right) \implies$

$- 8 a \left(\frac{1}{b} ^ \left(\textcolor{b l u e}{2} - \textcolor{red}{- 5}\right)\right) \implies$

$- 8 a \left(\frac{1}{b} ^ \left(\textcolor{b l u e}{2} + \textcolor{red}{5}\right)\right) \implies$

$- 8 a \left(\frac{1}{b} ^ 7\right) \implies$

$\frac{- 8 a}{b} ^ 7$

Jun 23, 2018

$- \frac{8 a}{{b}^{7}}$

#### Explanation:

$\frac{- 16 {a}^{-} 3 {b}^{-} 5}{2 {a}^{-} 4 {b}^{2}}$

Use the rule for exponents: ${a}^{-} n = \frac{1}{a} ^ n$

and the rule ${a}^{n} / {a}^{m} = {a}^{n - m}$

$\frac{- 16 {a}^{-} 3 {b}^{-} 5}{2 {a}^{-} 4 {b}^{2}}$

first let's simplify:

$\frac{- 16 \cdot {a}^{-} 3 \cdot {b}^{-} 5}{2 \cdot {a}^{-} 4 \cdot {b}^{2}}$

$- 8 \cdot {a}^{- 3 - \left(- 4\right)} \cdot {b}^{- 5 - 2}$

$- 8 \cdot {a}^{- 3 + 4} \cdot {b}^{- 7}$

$- 8 \cdot a \cdot {b}^{- 7}$

Now move the negative exponents:

$- \frac{8 a}{{b}^{7}}$

Jun 23, 2018

$\frac{- 16 {a}^{-} 3 {b}^{-} 5}{2 {a}^{-} 4 {b}^{2}}$

Group the like terms.

$= \frac{- 16}{2} \cdot \frac{{a}^{-} 3}{{a}^{-} 4} \cdot \frac{{b}^{-} 5}{{b}^{2}}$

Use the rule $\frac{{x}^{p}}{{x}^{q}} = {x}^{p - q}$

$= - 8 \cdot {a}^{- 3 + 4} \cdot {b}^{- 5 - 2}$

Simplify the exponents.

$= - 8 \cdot a \cdot {b}^{-} 7$

Use the rule ${x}^{-} n = \frac{1}{{x}^{n}}$ to write it with positive exponents.

$= - 8 a \cdot \frac{1}{{b}^{7}}$

Simplify.

$= - \frac{8 a}{{b}^{7}}$