Example 1:
$$ \textup{Find the square root of } 36$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 36\\ \hline
\llap{2~~~~} 18 \\ \hline
\llap{3~~~~} 9 \\ \hline
\llap{3~~~~} 3 \\ \hline
1
\end{array}
$$
$$ \sqrt{36} = \sqrt{2×2×3×3}$$
$$ = \sqrt{2^2×3^2}$$
$$ = \sqrt{(2×3)^2}$$
$$ = \sqrt{6^2}$$
$$ = 6$$
Example 2:
$$ \textup{Find the square root of } 225$$
Solution:
$$
\begin{array}{|l}
\llap{3~~~~} 225 \\ \hline
\llap{3~~~~} 75 \\ \hline
\llap{5~~~~} 25 \\ \hline
\llap{5~~~~} 5 \\ \hline
1
\end{array}
$$
$$ \sqrt{225} = \sqrt{3×3×5×5}$$
$$ = \sqrt{3^2×5^2}$$
$$ = \sqrt{(3×5)^2}$$
$$ = \sqrt{15^2}$$
$$ = 15$$
Example 1:
$$ \textup{Find the square root of } 324$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 324\\ \hline
\llap{2~~~~} 162 \\ \hline
\llap{3~~~~} 81 \\ \hline
\llap{3~~~~} 27 \\ \hline
\llap{3~~~~} 9 \\ \hline
\llap{3~~~~} 3 \\ \hline
1
\end{array}
$$
$$ \sqrt{324} = \sqrt{2×2×3×3×3×3}$$
$$ = \sqrt{2^2×3^2×3^2}$$
$$ = \sqrt{(2×3×3)^2}$$
$$ = \sqrt{18^2}$$
$$ = 18$$
Example 2:
$$ \textup{Find the square root of } 324$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 4356 \\ \hline
\llap{2~~~~} 2178 \\ \hline
\llap{3~~~~} 1089 \\ \hline
\llap{3~~~~} 363 \\ \hline
\llap{11~~~~} 121 \\ \hline
\llap{11~~~~} 11 \\ \hline
1
\end{array}
$$
$$ \sqrt{4356} = \sqrt{2×2×3×3×11×11}$$
$$ = \sqrt{2^2×3^2×11^2}$$
$$ = \sqrt{(2×3×11)^2}$$
$$ = \sqrt{66^2}$$
$$ = 66$$
Example 1:
$$ \textup{Find the square root of } 324$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 15876\\ \hline
\llap{2~~~~} 7938 \\ \hline
\llap{3~~~~} 3969 \\ \hline
\llap{3~~~~} 1323 \\ \hline
\llap{3~~~~} 441\\ \hline
\llap{3~~~~} 147 \\ \hline
\llap{7~~~~} 49 \\ \hline
\llap{7~~~~} 7 \\ \hline
1
\end{array}
$$
$$ \sqrt{324} = \sqrt{2×2×3×3×3×3×7×7}$$
$$ = \sqrt{2^2×3^2×3^2×7^2}$$
$$ = \sqrt{(2×3×3×7)^2}$$
$$ = \sqrt{126^2}$$
$$ = 126$$
Example 2:
$$ \textup{Find the square root of } 324$$
Solution:
$$
\begin{array}{|l}
\llap{2~~~~} 1296 \\ \hline
\llap{2~~~~} 648 \\ \hline
\llap{2~~~~} 324 \\ \hline
\llap{2~~~~} 162 \\ \hline
\llap{3~~~~} 81\\ \hline
\llap{3~~~~} 27 \\ \hline
\llap{3~~~~} 9 \\ \hline
\llap{3~~~~} 3 \\ \hline
1
\end{array}
$$
$$ \sqrt{4356} = \sqrt{2×2××2×2×3×3×3×3}$$
$$ = \sqrt{2^2×2^2×3^2×3^2}$$
$$ = \sqrt{(2×2×3×3)^2}$$
$$ = \sqrt{36^2}$$
$$ = 36$$