Square root definition

In this chapter we will learn about the concept of square root and the method to calculate it.

What does square root mean ?

Let ” a ” be the positive integer.

On squaring the number, we get \mathtt{a^{2}} .

Now, if we take square root of number \mathtt{a^{2}} , we will get back the original number ” a “

Hence, square root is the inverse operation of squaring the number.

How to represent square root of number ?

If you want to represent square root of any number, place ” \mathtt{\sqrt{}} ” sign above that number.

For example;

square root of 16 ⟹ \mathtt{\sqrt{16}}

square root of 25 ⟹ \mathtt{\sqrt{25}}

How to find square root of number ?

Here we will learn to find square root of perfect square.

(a) Do the prime factorization of given number

(b) Arrange all the factor in pair of two.

This means that all the factors will have exponent of 2.

(c) Remove all the exponents and simply multiply the factors to get the square root.

I hope you understood the above concept, let us now solve some problems.

Example 01
Find the square root of 100

Solution

(a) Do the prime factorization.

(b) Arrange all factor in pair of two.

The above factorization can be expressed as;

\mathtt{100\ =2^{2} \times 5^{2}}

All the factors have exponent 2 means that the given number is a perfect square.

(c) Remove the exponent and multiply the numbers to get square root.

Square root 100 ⟹ 2 x 5

Square root 100 ⟹ 10

Hence, 10 is the square root of 100.

It means that if we multiply 10 by itself we get 100.

Example 02
Find the square root of 256

Solution

(a) Do the prime factorization.

(b) Arrange all factor in pair of two.

The above factorization can be expressed as;

\mathtt{256\ =2^{2} \times 2^{2} \times 2^{2} \times 2^{2}}

All the factors have power 2, this mean that the above number is perfect square.

(c) To get the square root, remove the exponents and multiply the numbers.

Square root 256 ⟹ 2 x 2 x 2 x 2

Square root 256 ⟹ 16

Hence, 16 is the square root of number 256.

Example 03
Find the square root of 324

Solution

(a) Do the prime factorization of number.

(b) Now form a pair of two of all the factors.

The prime factorization can be represented as;

\mathtt{324\ =2^{2} \times 3^{2} \times 3^{2}}

Since all the factors have exponent of two, the number 324 is a perfect square.

(c) To get the square root, remove the exponents and multiply the numbers.

Square root 324 ⟹ 2 x 3 x 3

Square root 324 ⟹ 18

Hence, 18 is the square root of number 324.

This mean that if we multiply 18 by itself, we get 324.

Example 04
Find the square root of 2025

Solution
(a) Do the prime factorization

(b) Arrange the factors in pair of two

The above factorization can be expressed as;

\mathtt{2025\ =3^{2} \times 3^{2} \times 5^{2}}

Since, all the factors have exponent two, the number 2025 is a perfect square.

(c) To get the square root, remove the exponents and multiply the number.

Square root 2025 ⟹ 3 x 3 x 5

Square root 2025 ⟹ 45

Hence, 45 is the square root of number 2025.

Example 05
Find the square root of 288

Solution

(a) Do the prime factorization.

The above factorization can be represented as;

\mathtt{288\ =2^{2} \times 2^{2} \times 3^{2} \times 2}

Here, the factor 2 at the end has exponent 1.

From the above factorization, we can see that when we form pair of two, the factor 2 is left out. Hence, the number is not a perfect square.

We cannot calculate the square root of non-perfect square using this method.