# How do you simplify root3(216,000)?

Jan 11, 2016

$\sqrt[3]{216 , 000} = 60$

#### Explanation:

Extracting the most obvious cube:
$\textcolor{w h i t e}{\text{XXX}} 216 , 000 = 216 \times {10}^{3}$

Then factor out basic primes until we get to a complete factoring:
$\textcolor{w h i t e}{\text{XXX}} = 2 \times 108 \times {10}^{3}$

$\textcolor{w h i t e}{\text{XXX}} = {2}^{2} \times 54 \times {10}^{3}$

$\textcolor{w h i t e}{\text{XXX}} = {2}^{3} \times 27 \times {10}^{3}$

$\textcolor{w h i t e}{\text{XXX}} = {2}^{3} \times 3 \times 9 \times {10}^{3}$

$\textcolor{w h i t e}{\text{XXX}} = {2}^{3} \times {3}^{3} \times {10}^{3}$
$\textcolor{w h i t e}{\text{XXXXXX}}$In practice we probably would have recognized the cube factors before going this far.

Therefore
$\textcolor{w h i t e}{\text{XXX}} \sqrt[3]{216 , 000} = \sqrt[3]{{2}^{3} \times {3}^{3} \times {10}^{3}} = 2 \times 3 \times 10 = 60$