Question

What are the methods to find the square root of a number?

Answer 1

Step 1. Definition of the square root of a number.

A square root of a number is a number that produces the same result when multiplied by itself.

The square root of any number $x$ is $\sqrt{x}$.

Example:

$\begin{array}{rcl}8\times 8& =& 64\\ & \Rightarrow & \sqrt{64}=8\end{array}$

Step 2. Methods for calculating a number's square root.

(i) Prime factorization method:
Prime factorization is a way to determine the prime factors of a given number as follows:

Example,

$\begin{array}{rcl}144& =& 2\times 2\times 2\times 2\times 3\times 3\\ & \Rightarrow & \sqrt{144}=\sqrt{\left(2\times 2\right)\times \left(2\times 2\right)\times \left(3\times 3\right)}\\ & \Rightarrow & \sqrt{144}=2\times 2\times 3\\ & \Rightarrow & \sqrt{144}=12\end{array}$

Thus, the square root of $144$ is $12$.

Repeated subtraction method:

Subtract consecutive odd numbers from the provided number until the difference is zero. The number of times subtraction was carried out to obtain $0$ gives the square root of the number.

Consider the following example to find the square root of $16$ using the repeated subtraction method.

$$\begin{gathered} 16-1=15 \\ 15-3=12 \\ 12-5=7 \\ 7-7=0 \end{gathered}$$

In four-step, we get zero.

Thus, the square root of $16$ will be $4$.

Long division method:

Consider the following example to find the square root of $83$ using the long division method.

The square roots of $83$ is $9.11$.

Hence, the methods for finding the square roots are prime factorization, repeated subtraction method, and long division method.