Flexi Says:
The square root of a square number can be found using the prime factorization method. To find the square root of a given square number by prime factorization, we follow the following steps:
- Obtain the prime factorization of the given natural number.
- Make pairs of identical factors.
- Take one factor from each pair and find their product. The product so obtained is the square root of the given number.
For example:
Find the square root of 509796 by prime factorisation.
By prime factorisation, we get@$$\begin{align*} \begin{array}{|l} \llap{2~~~~} 509796 \\ \hline \llap{2~~~~} 254898 \\ \hline \llap{3~~~~} 127449 \\ \hline \llap{3~~~~} 42483 \\ \hline \llap{7~~~~} 14161 \\ \hline \llap{7~~~~} 2023 \\ \hline \llap{17~~~~} 289 \\ \hline\llap{17~~~~} 17 \\ \hline 1 \end{array}\end{align*}@$$
@$$\begin{align*}509796 = \underbrace{2 \times 2} \times \underbrace{3 \times 3} \times \underbrace{7 \times 7} \times \underbrace{17 \times 17}\end{align*}@$$
Taking one factor from each pair after grouping the factors into pairs of identical factors, we get@$$\begin{align*}\sqrt{509796} = 2 \times 3 \times 7 \times 17 = 714\end{align*}@$$
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