# How do you simplify \frac { - 18p ^ { - 2} r ^ { 3} h ^ { - 8} v ^ { - 8} } { 9a ^ { - 1} p ^ { 3} r ^ { 3} v ^ { - 2} w }?

Apr 17, 2018

$= - \frac{2 a}{{p}^{5} {h}^{8} {v}^{6} w}$

#### Explanation:

$\setminus \frac{- 18 {p}^{- 2} {r}^{3} {h}^{- 8} {v}^{- 8}}{9 {a}^{- 1} {p}^{3} {r}^{3} {v}^{- 2} w}$

First, let's separate the like-terms to make it easier to see what's going on here:

$= \frac{- 18}{9} \cdot \frac{1}{{a}^{- 1}} \cdot {p}^{-} \frac{2}{p} ^ 3 \cdot {r}^{3} / {r}^{3} \cdot {h}^{-} \frac{8}{1} \cdot {v}^{-} \frac{8}{v} ^ - 2 \cdot \frac{1}{w}$

$= \frac{- 18}{9} \cdot {a}^{0} / \left({a}^{- 1}\right) \cdot {p}^{-} \frac{2}{p} ^ 3 \cdot {r}^{3} / {r}^{3} \cdot {h}^{-} \frac{8}{h} ^ 0 \cdot {v}^{-} \frac{8}{v} ^ - 2 \cdot {w}^{0} / w$

since anything to the $0$th power is $1$.

The rules for like-terms and powers is:

• Multiplication of terms requires addition of powers
• Division of terms requires subtraction of powers

We will need to use the latter.

$= - 2 \cdot {a}^{0 - \left(- 1\right)} \cdot {p}^{- 2 - 3} \cdot {r}^{3 - 3} \cdot {h}^{- 8 - 0} \cdot {v}^{- 8 - \left(- 2\right)} \cdot {w}^{0 - 1}$

$= - 2 \cdot {a}^{1} \cdot {p}^{- 5} \cdot {r}^{0} \cdot {h}^{- 8} \cdot {v}^{- 6} \cdot {w}^{- 1}$

$= - 2 a {p}^{-} 5 {h}^{-} 8 {v}^{-} 6 {w}^{-} 1$

Rewriting with only positive powers (if you would like to):

$= - \frac{2 a}{{p}^{5} {h}^{8} {v}^{6} w}$

Apr 17, 2018

$- \frac{2 a}{{p}^{5} {h}^{8} {v}^{6} w}$ or $- 2 a {p}^{-} 5 {h}^{-} 8 {v}^{-} 6 {w}^{-} 1$

#### Explanation:

first, you can find individual quotients:

$\frac{18}{9} = - 2$

$\frac{1}{{a}^{-} 1} = {a}^{1} = a$

${p}^{-} \frac{2}{p} ^ 2 = {p}^{- 2 - 3} = {p}^{-} 5$

${r}^{3} / {r}^{3} = {r}^{3 - 3} = {r}^{0} = 1$

${h}^{-} \frac{8}{1} = {h}^{-} 8$

${v}^{-} \frac{8}{v} ^ - 2 = {v}^{- 8 - - 2} = {v}^{- 8 + 2} = {v}^{-} 6$

$\frac{1}{w} = {w}^{-} 1$

multiplying all of these together gives a simplified version of the fraction.

hence, the expression can be written as $- 2 a {p}^{-} 5 {h}^{-} 8 {v}^{-} 6 {w}^{-} 1$.

if you want to have the numerator as only positive exponents, you can write the negative exponents in the previous expression in the denominator.

this gives $- \frac{2 a}{{p}^{5} {h}^{8} {v}^{6} w}$.