What is the square root of 464?

Hint: Initially, find the factors of 464 by using prime factorisation. We know that prime factorisation is a method to find the prime factors of a given number, say a composite number. Let us consider this as equation (1). Now by applying the square root on both sides we can have the value of square root of 464.

Complete step-by-step answer:
From the question, it is clear that we have to find the square root of 464.
First by using prime factorisation, we should simply the given number 464 such that it will be easy to find the square root of 464.
\[\begin{align}
  & 2\left| \!{\underline {\,
  464 \,}} \right. \\
 & 2\left| \!{\underline {\,
  232 \,}} \right. \\
 & 2\left| \!{\underline {\,
  116 \,}} \right. \\
 & 2\left| \!{\underline {\,
  58 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  29 \,}} \right. \\
\end{align}\]
So, we can write
\[\begin{align}
  & \Rightarrow 464=2\times 2\times 2\times 2\times 29 \\
 & \Rightarrow 464={{2}^{4}}\times 29 \\
\end{align}\]
So, now 464 is shown in the form of prime factors like above.
Let us consider
\[464={{2}^{4}}\times 29......(1)\]
Now apply square root on both sides, then we get
\[\begin{align}
  & \Rightarrow \sqrt{464}=\sqrt{{{2}^{4}}\times 29} \\
 & \Rightarrow \sqrt{464}={{2}^{2}}\sqrt{29} \\
 & \Rightarrow \sqrt{464}=4\sqrt{29} \\
\end{align}\]
Let us consider this as equation (2).
\[\sqrt{464}=4\sqrt{29}.....(2)\]
We know that the square root of 29 is equal to 5.385.
Now apply this in equation (2), then
So, we get
\[\sqrt{464}=4(5.385)=21.54\]
So, it is clear that the square root of 464 is equal to 21.54.

Note: Students may undergo calculation mistakes while solving this problem. If a small mistake is done, then the final answer may get interrupted. So, solving the solutions in a careful manner is important. Students should also be aware of prime factorisation to solve the problem.