# How do you simplify (20qr^-2t^-5)/(rq^0r^4t^-2)?

May 15, 2017

See a solution process below:

#### Explanation:

First, rewrite this expression as:

$20 \left(\frac{q}{q} ^ 0\right) \left({r}^{-} \frac{2}{r \cdot {r}^{4}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right)$

Next, simplify the $q$ term using this rule of exponents:

${a}^{\textcolor{red}{0}} = 1$

$20 \left(\frac{q}{q} ^ \textcolor{red}{0}\right) \left({r}^{-} \frac{2}{r \cdot {r}^{4}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies 20 \left(\frac{q}{\textcolor{red}{1}}\right) \left({r}^{-} \frac{2}{r \cdot {r}^{4}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies$

$20 q \left({r}^{-} \frac{2}{r \cdot {r}^{4}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right)$

Next, use these rules of exponents to simplify the $r$ terms:

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

$20 q \left({r}^{-} \frac{2}{r \cdot {r}^{4}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies 20 q \left({r}^{-} \frac{2}{{r}^{\textcolor{red}{1}} \cdot {r}^{\textcolor{b l u e}{4}}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies$

$20 q \left({r}^{-} \frac{2}{{r}^{\textcolor{red}{1} + \textcolor{b l u e}{4}}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies 20 q \left({r}^{\textcolor{red}{- 2}} / \left({r}^{\textcolor{b l u e}{5}}\right)\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies$

$20 q \left(\frac{1}{{r}^{\textcolor{b l u e}{5} - \textcolor{red}{- 2}}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies 20 q \left(\frac{1}{{r}^{\textcolor{b l u e}{5} + \textcolor{red}{2}}}\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies$

$20 q \left(\frac{1}{r} ^ 7\right) \left({t}^{-} \frac{5}{t} ^ - 2\right) \implies \frac{20 q}{r} ^ 7 \left({t}^{-} \frac{5}{t} ^ - 2\right)$

Now, use this rule to simplify the $t$ terms:

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

$\frac{20 q}{r} ^ 7 \left({t}^{\textcolor{red}{- 5}} / {t}^{\textcolor{b l u e}{- 2}}\right) \implies \frac{20 q}{r} ^ 7 \left(\frac{1}{t} ^ \left(\textcolor{b l u e}{- 2} - \textcolor{red}{- 5}\right)\right) \implies \frac{20 q}{r} ^ 7 \left(\frac{1}{t} ^ \left(\textcolor{b l u e}{- 2} + \textcolor{red}{5}\right)\right) \implies$

$\frac{20 q}{r} ^ 7 \left(\frac{1}{t} ^ 3\right) \implies \frac{20 q}{{r}^{7} {t}^{3}}$