# How do you simplify ((3x^7)/2)^6(x^2)^6?

Jun 20, 2018

color(purple)(=> (729/64)x^54

#### Explanation:

${a}^{m} \cdot {a}^{n} = {a}^{m + n} , {\left({a}^{m}\right)}^{n} = {a}^{m n}$

${\left(\frac{3 {x}^{7}}{2}\right)}^{6} \cdot {\left({x}^{2}\right)}^{6}$

$\implies {\left(\left(\frac{3 {x}^{7}}{2}\right) \cdot {x}^{2}\right)}^{6}$

$\implies {\left(\frac{3 {x}^{9}}{2}\right)}^{6}$

$\implies \frac{{3}^{6} \cdot {x}^{9 \cdot 6}}{2} ^ 6$

color(purple)(=> (729/64)x^54

Jun 20, 2018

$\frac{729}{64} {x}^{54}$

#### Explanation:

In general ${\textcolor{red}{a}}^{\textcolor{\lim e}{p}} \cdot {\textcolor{b l u e}{b}}^{\textcolor{\lim e}{p}} = {\left(\textcolor{red}{a} \cdot \textcolor{b l u e}{b}\right)}^{\textcolor{\lim e}{p}}$

So
$\textcolor{w h i t e}{\text{XXX}} {\left(\textcolor{red}{\frac{3 {x}^{7}}{2}}\right)}^{\textcolor{\lim e}{6}} {\left(\textcolor{b l u e}{{x}^{2}}\right)}^{\textcolor{\lim e}{6}} = {\left(\frac{\textcolor{red}{3 {x}^{7}} \cdot \textcolor{b l u e}{{x}^{2}}}{\textcolor{red}{2}}\right)}^{\textcolor{\lim e}{6}}$

$\textcolor{w h i t e}{\text{XXX}} = {\left(\frac{3 {x}^{9}}{\textcolor{red}{2}}\right)}^{\textcolor{\lim e}{6}}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{{3}^{6}}{{2}^{6}} {x}^{9 \cdot 6}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{729}{64} {x}^{54}$