# How do you simplify ((2r^3)/s)^2(s/r^3)^3 and write it using only positive exponents?

Feb 7, 2017

${\left(\frac{2 {r}^{3}}{s}\right)}^{2} {\left(\frac{s}{r} ^ 3\right)}^{3} = \frac{4 s}{{r}^{3}}$

#### Explanation:

${\left(\frac{2 {r}^{3}}{s}\right)}^{2} {\left(\frac{s}{r} ^ 3\right)}^{3} = \frac{{2}^{2} {r}^{3 \cdot 2}}{{s}^{2}} \cdot {s}^{3} / {r}^{3 \cdot 3}$

$= \frac{4 {r}^{6}}{s} ^ 2 \cdot {s}^{3} / {r}^{9}$

$= \frac{4 {s}^{3 - 2}}{{r}^{9 - 6}}$

$= \frac{4 s}{{r}^{3}}$

Feb 7, 2017

See the entire simplification process below:

#### Explanation:

First use these two rules for exponents to begin simplifying:

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

${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${\left(\frac{2 {r}^{3}}{s}\right)}^{2} {\left(\frac{s}{r} ^ 3\right)}^{3} \to {\left(\frac{{2}^{\textcolor{red}{1}} {r}^{\textcolor{red}{3}}}{s} ^ \textcolor{red}{1}\right)}^{\textcolor{b l u e}{2}} {\left({s}^{\textcolor{red}{1}} / {r}^{\textcolor{red}{3}}\right)}^{\textcolor{b l u e}{3}} \to$

$\frac{{2}^{\textcolor{red}{1} \times \textcolor{b l u e}{2}} {r}^{\textcolor{red}{3} \times \textcolor{b l u e}{2}}}{s} ^ \left(\textcolor{red}{1} \times \textcolor{b l u e}{2}\right) \times {s}^{\textcolor{red}{1} \times \textcolor{b l u e}{3}} / {r}^{\textcolor{red}{3} \times \textcolor{b l u e}{3}} \to \frac{{2}^{2} {r}^{6} {s}^{3}}{{s}^{2} {r}^{9}} \to \frac{4 {r}^{6} {s}^{3}}{{s}^{2} {r}^{9}}$

Now, use these two rules for exponents to complete the simplification:

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

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

$\frac{4 {r}^{\textcolor{red}{6}} {s}^{\textcolor{red}{3}}}{{s}^{\textcolor{b l u e}{2}} {r}^{\textcolor{b l u e}{9}}} \to \frac{4 {s}^{\textcolor{red}{3} - \textcolor{b l u e}{2}}}{r} ^ \left(\textcolor{b l u e}{9} - \textcolor{red}{6}\right) \to \frac{4 {s}^{1}}{r} ^ 3 \to \frac{4 s}{r} ^ 3$