Question
Find the square root of the following by prime factorization.$27225$
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Answer
Resolving $27225$ into prime factors, $27225 = 3 \times 3 \times 5 \times 5 \times 11 \times 11$
$\begin{array}{c|c}3&27225 \\ \hline 3 & 9075 \\\hline 5&3025\\\hline5&605 \\\hline11&121\\\hline11&11\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $27225 = (3 \times 3) \times (5 \times 5) \times (11 \times 11)$
Taking one factor for each pair, we get the square root of $27225 3 \times 5 \times 11 = 165$
$\begin{array}{c|c}3&27225 \\ \hline 3 & 9075 \\\hline 5&3025\\\hline5&605 \\\hline11&121\\\hline11&11\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $27225 = (3 \times 3) \times (5 \times 5) \times (11 \times 11)$
Taking one factor for each pair, we get the square root of $27225 3 \times 5 \times 11 = 165$
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