Example 1:
$$ \textup{Find the cube root of } 216$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 216\\ \hline
\llap{2~~~~} 108 \\ \hline
\llap{2~~~~} 54 \\ \hline
\llap{3~~~~} 27 \\ \hline
\llap{3~~~~} 9 \\ \hline
\llap{3~~~~} 3 \\ \hline
1
\end{array}
$$
$$ \sqrt[3]{216} = \sqrt[3]{2×2×2×3×3×3}$$
$$ = \sqrt[3]{2^3×3^3}$$
$$ = \sqrt[3]{(2×3)^3}$$
$$ = \sqrt[3]{6^3}$$
$$ = 6$$
Example 2:
$$ \textup{Find the cube root of } 1000$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 1000 \\ \hline
\llap{2~~~~} 500 \\ \hline
\llap{2~~~~} 250 \\ \hline
\llap{5~~~~} 125 \\ \hline
\llap{5~~~~} 25 \\ \hline
\llap{5~~~~} 5 \\ \hline
1
\end{array}
$$
$$ \sqrt[3]{1000} = \sqrt[3]{2×2×2×5×5×5}$$
$$ = \sqrt[3]{2^3×5^3}$$
$$ = \sqrt[3]{(2×5)^3}$$
$$ = \sqrt[3]{10^3}$$
$$ = 10$$
Example 1:
$$ \textup{Find the cube root of } 1728$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 1728 \\ \hline
\llap{2~~~~} 864 \\ \hline
\llap{2~~~~} 432 \\ \hline
\llap{2~~~~} 216 \\ \hline
\llap{2~~~~} 108 \\ \hline
\llap{2~~~~} 54 \\ \hline
\llap{3~~~~} 27 \\ \hline
\llap{3~~~~} 9 \\ \hline
\llap{3~~~~} 3 \\ \hline
1
\end{array}
$$
$$ \sqrt[3]{324} = \sqrt[3]{2×2×2×2×2×2×3×3×3}$$
$$ = \sqrt[3]{2^3×2^3×3^3}$$
$$ = \sqrt[3]{(2×2×3)^3}$$
$$ = \sqrt[3]{12^3}$$
$$ = 12$$
Example 2:
$$ \textup{Find the cube root of } 74088$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 74088 \\ \hline
\llap{2~~~~} 37044 \\ \hline
\llap{2~~~~} 18522 \\ \hline
\llap{3~~~~} 9261 \\ \hline
\llap{3~~~~} 3087 \\ \hline
\llap{3~~~~} 1029 \\ \hline
\llap{7~~~~} 343 \\ \hline
\llap{7~~~~} 49 \\ \hline
\llap{7~~~~} 7 \\ \hline
1
\end{array}
$$
$$ \sqrt[3]{4356} = \sqrt[3]{2×2×2×3×3×3×7×7×7}$$
$$ = \sqrt[3]{2^3×3^3×7^3}$$
$$ = \sqrt[3]{(2×3×7)^3}$$
$$ = \sqrt[3]{42^3}$$
$$ = 42$$