Question

Find the square root of $441$ by prime factorization method.

Answer 1

Finding the square root

$$\begin{gathered} 441=3\times 3\times 7\times 7 \\ \Rightarrow 441=3^2\times 7^2 \\ \Rightarrow {\left(441\right)}^{\frac{1}{2}}={\left(3^2\times 7^2\right)}^{\frac{1}{2}} \\ \Rightarrow {\left(441\right)}^{\frac{1}{2}}={\left({21}^2\right)}^{\frac{1}{2}} \\ \Rightarrow {\left(441\right)}^{\frac{1}{2}}={21}^{\frac{2}{2}}[\because {\left(a^m\right)}^n=a^{mn}] \\ \Rightarrow {\left(441\right)}^{\frac{1}{2}}=21 \end{gathered}$$

Hence, the square root of $441$ is $21$