Question

Find the square root of $4096$ by prime factorization method.

Answer 1

Finding the square root

Since prime factorization of $4096$ as

$4096=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$

Therefore,

$$\begin{gathered} 4096=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 \\ \Rightarrow 4096=2^{12} \\ \Rightarrow {\left(4096\right)}^{\frac{1}{2}}={\left(2^{12}\right)}^{\frac{1}{2}} \\ \Rightarrow {\left(4096\right)}^{\frac{1}{2}}=2^{\frac{12}{2}}[\because {\left(a^m\right)}^n=a^{mn}] \\ \Rightarrow {\left(4096\right)}^{\frac{1}{2}}=2^6 \\ \Rightarrow {\left(4096\right)}^{\frac{1}{2}}=64 \end{gathered}$$

$\therefore$The square root of $4096$ is $64$

Hence, the square root of $4096$ is $64$