Question

How do you find the square root of $324$ ?

Answer 1

Hint:
For finding the square root of $324$ we need to break down this number into the product of numbers by doing the prime factor. Then we will solve this by making a pair if it will be otherwise we will write the same as we got from the factor.

Complete Step by Step Solution:
So the prime breakdown of the number $324$ will be
$ \Rightarrow 324 = 2 \times 162$
Again doing the breakdown of $162$ , we will get
$ \Rightarrow 324 = 2 \times 2 \times 81$
Again doing the breakdown of $81$ , we will get
$ \Rightarrow 324 = 2 \times 2 \times 3 \times 27$
Similarly, we will do this breakdown till possible and finally, we will get the number
$ \Rightarrow 324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$
So, now from the question, we can write it as
\[ \Rightarrow \sqrt {324} = \sqrt {\left( {2 \times 2} \right)\left( {3 \times 3} \right)\left( {3 \times 3} \right)} \]
The right side of the equation can be written as
$ \Rightarrow \sqrt {324} = 2 \times 3 \times 3$
Again the right side of the equation on solving it can be written as
$ \Rightarrow \sqrt {324} = 18$

therefore, the square root of $324$ will be $18$.

Additional information:
The square root of a number is a number when multiplied by itself equals the whole number. Some of the numbers don’t have the square root.

Note:
This can be solved by using the shorter way too. Only we need to break it into such a way that there should be a square of it. Like we can write the above question as
$ \Rightarrow 324 = 2 \times 162$
And taking the RHS, we will split both the root so we get
$ \Rightarrow 324 = 2 \times 2 \times 81$
Taking root both sides, we get
$ \Rightarrow \sqrt {324} = \sqrt {2 \times 2 \times 81} $
And as we know that $\sqrt {81} = 9$ , hence by substituting this value, we will get the equation as
$ \Rightarrow \sqrt {324} = 9\sqrt {2 \times 2} $
And on solving, we get
$ \Rightarrow \sqrt {324} = 18$
So, in this way also we can solve such types of questions.