Question

Find the square root by prime factorization: $24336$

Answer 1

Hint: Square root of a number is a value, which when multiplied by itself gives the original number. Suppose, ‘x’ is the square root of ‘y’, then it is represented as $x = \sqrt y $ or we can express the same equation as ${x^2} = y$ . Here we have to calculate the square root of the number $24336$. Now, to simplify the square root of $24336$, we first do the prime factorization of the number and take the factors occurring in pairs outside of the square root radical.

Complete step by step answer:
So, we have $24336$.
 We will first find out all the prime factors of the number using the method of prime factorisation.
So, we get,
\[\begin{align}
  & 2\left| \!{\underline {\,
  24336 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12168 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6084 \,}} \right. \\
 & 2\left| \!{\underline {\,
  3042 \,}} \right. \\
 & 3\left| \!{\underline {\,
  1521\,}} \right. \\
& 3\left| \!{\underline {\,
  507\,}} \right. \\
 & 13\left| \!{\underline {\,
  169\,}} \right. \\
 & 13\left| \!{\underline {\,
  13\,}} \right. \\
\end{align}\]
Hence, $24336$ can be factored as,
$24336 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 13 \times 13$
Now, expressing the prime factorization in powers and exponents, we get,
$24336 = {2^4} \times {3^2} \times {13^2}$
We can see that $2$ is multiplied four times and hence the power of $2$ is four. Similarly, $13$ is multiplied twice and the power is two. Also, $3$ is multiplied twice and power of $3$ is two.
Now, taking square root, we get,
$\sqrt {24336} = \sqrt {{2^4} \times {3^2} \times {{13}^2}} $
Since we know that ${2^4}$, ${3^2}$, and ${13^2}$ are perfect squares. So, we can take this outside of the square root. So, we have,
So, $\sqrt {24336} = {2^2} \times 3 \times 13$
Simplifying further, we get,
$ \Rightarrow \sqrt {24336} = 4 \times 39$
$ \Rightarrow \sqrt {24336} = 156$
Hence, square root of $24336$ is $156$.

Note:
Here $\sqrt {} $ is the radical symbol used to represent the root of numbers. The number under the radical symbol is called radicand. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number. To find the factors, find the smallest prime number that divides the given number and divide it by that number, and then again find the smallest prime number that divides the number obtained and so on. The set of prime numbers obtained that are multiplied to each other to form the bigger number are called the factors.