# Finding Square Root of a Number by Prime Factorization

Area of a square is the product of its sides. In a square, all the sides have the same length. Hence, if one of the sides is known then the area can be easily calculated. Area of square = \(side~ × ~side\) = \(side^2\). Now what if area of the square is given and we have to calculate the length of its side? This can easily be done by finding the square root of the area. The value obtained after calculating the square root will be the length of its side. For example, the area of a square is 225. Its length will be:

\(225\) = \(a^2\)

\(\Rightarrow~15~ ×~ 15\) = \(a^2\)

\(\Rightarrow a\) = \(15\)

Therefore the side of the square is 15 units.

### Different ways of finding a number’s square root

Subtaction once inversed would become addition and divivsion once inversed becomes multiplication. In this way, a square number’s inverse would be a square root. For instance,

\(1^2\) = \(1\), the square root of \(1\) is \(1\)

\(4^2\) = \(16\), square root of \(16\) is \(4\) and so on.

Students should note that positive valued square roots of the natural numbers would be taken into consideration.

Prime Factorization

Let us consider the prime factors of a number and its square. For example 12 and 144,

12 = 2 × 2 × 3

144 = 2 × 2 × 3 × 2 × 2 × 3

It can be observed that the prime factors in the prime factorization of a square number occurs twice the number of times it occurs in the number itself. For example, finding the prime factors of 576.

For finding the square root, firstly we have to pair the common factors.

\(576\) = \(\underline{2~ ×~ 2} ~×~\underline{ 2 ~×~ 2} ~×~ \underline{2~ ×~ 2} ~×~\underline{ 3~ ×~ 3}\)

\(\Rightarrow~ 576\) = \(2^2~ × ~2^2 ~× ~2^2~ ×~ 3^2\)

The square root of 576 will be:

\(2 ~× ~2 ~×~ 2~ ×~ 3\) = \(24\).