# How do you multiply 9th root of 9x squared times 6th root of 27 x raise to 5?

Jun 28, 2018

The 27th root of 729x raised to the 5th power.

#### Explanation:

9th root of 9x squared can be written as
$\textcolor{w h i t e}{\text{XXX}} {\left(9 x\right)}^{\frac{2}{9}} = {\left({3}^{2}\right)}^{\frac{2}{9}} \cdot {x}^{\frac{2}{9}}$
$\textcolor{w h i t e}{\text{XXXXXX}} = {3}^{\frac{4}{9}} \cdot {x}^{\frac{2}{9}}$

6th root of 27x raised to 5 can be written as
$\textcolor{w h i t e}{\text{XXX}} {\left(27 x\right)}^{\frac{5}{6}} = {\left({3}^{3}\right)}^{\frac{5}{6}} \cdot {x}^{\frac{5}{6}}$
$\textcolor{w h i t e}{\text{XXXXXXX}} = {3}^{\frac{15}{6}} \cdot {x}^{\frac{5}{6}}$
$\textcolor{w h i t e}{\text{XXXXXXX}} = {3}^{\frac{5}{2}} \cdot {x}^{\frac{5}{6}}$

${\left(9 x\right)}^{\frac{2}{9}} \times {\left(27 x\right)}^{\frac{5}{6}}$

$\textcolor{w h i t e}{\text{XXX}} = {3}^{\frac{4}{9}} \cdot {x}^{\frac{2}{9}} \times {3}^{\frac{5}{2}} \cdot {x}^{\frac{5}{6}}$

$\textcolor{w h i t e}{\text{XXX}} = {3}^{\frac{4}{9}} \cdot {3}^{\frac{5}{2}} \times {x}^{\frac{2}{9}} \cdot {x}^{\frac{5}{6}}$

$\textcolor{w h i t e}{\text{XXX}} = {3}^{\frac{10}{9}} \cdot {x}^{\frac{5}{27}}$

$\textcolor{w h i t e}{\text{XXX}} = {3}^{\frac{30}{27}} \cdot {x}^{\frac{5}{27}}$

$\textcolor{w h i t e}{\text{XXX}} = {\left({3}^{6}\right)}^{\frac{5}{27}} \cdot {x}^{\frac{5}{27}}$

$\textcolor{w h i t e}{\text{XXX}} = {729}^{\frac{5}{27}} \cdot {x}^{\frac{5}{27}}$

$\textcolor{w h i t e}{\text{XXX}} = {\left(729 x\right)}^{\frac{5}{27}}$