Question

Find the square root of $8649$ using the Prime factorization method?

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 $. To solve this problem, we first do the prime factorization of the number $8649$ and then multiply the factors by taking factors appearing in pairs only once.

Complete step by step solution:
In the given problem, we have to calculate the value of the square root of $8649$.
So, we do the prime factorization of the number. So, we have,
\[\begin{align}
  & 3\left| \!{\underline {\,
  8649 \,}} \right. \\
 & 3\left| \!{\underline {\,
  2883 \,}} \right. \\
 & 31\left| \!{\underline {\,
  961 \,}} \right. \\
 & 31\left| \!{\underline {\,
  31 \,}} \right. \\
\end{align}\]
So, we have the factors of the number $8649$ as: $8649 = 3 \times 3 \times 31 \times 31$
Expressing the product of prime factors in form of powers and exponents, we get
 $ \Rightarrow 8649 = {3^2} \times {31^2}$
Now, we get the square root of $8649$ as,
$ \Rightarrow \sqrt {8649} = \sqrt {{3^2} \times {{31}^2}} $
Now, we know that ${\left( 3 \right)^2}$ and ${\left( {31} \right)^2}$ are perfect squares. So, we can take them out of the square root. Simplifying the expression, we get,
$ \Rightarrow \sqrt {8649} = 3 \times 31$
Simplifying the calculations, we get,
$ \Rightarrow \sqrt {8649} = 93$
Hence, the square root of $8649$ is $93$.

Note:
Here $\sqrt{\text{ }}$ 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.