# Perfect Square

In this chapter we will learn the concept the perfect square along with method of calculation.

## What is Perfect Square ?

A perfect square is formed when an integer is multiplied by itself .

Hence, we we square a integer, the resultant number is a perfect square.

For example;
Let’s multiply the number 6 with itself.

\mathtt{\Longrightarrow \ 6\ \times \ 6}\\\ \\ \mathtt{\Longrightarrow \ 36}

Hence, number 36 is a perfect square since it is formed by square of number 6.

Example 02
Multiply number 12 by itself

\mathtt{\Longrightarrow \ 12\ \times \ 12}\\\ \\ \mathtt{\Longrightarrow \ 144}

Here number 144 is perfect square since it is formed by squaring number 12.

## Check if number is perfect square or not

Suppose you are provided with number 16.

Now in order to check if the number is perfect square, follow the below steps;

(i) Do prime factorization of number

(ii) If all the factors are arranged in pair of two then the number is perfect square.

In this case, all the factors have exponent two above it.

(iii) Now remove all the exponent and simply multiply the factor to get the square root.

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

Example 01
Check if number 81 is perfect square.

Solution

(a) Do Prime factorization of 81

(b) Check if all the factors are forming pair of two.

The above prime factorization can be represented as;

\mathtt{81\ =3^{2} \times 3^{2}}

Since, all the factors have exponent 2, it means all the factors can be arranged in pair of 2.

Hence, number 81 is a perfect square.

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

Square root of 81 ⟹ 3 x 3

Square root of 81 ⟹ 9

Hence, 9 is the square root of number 81.

This means that it we multiply 9 by itself we get 81.

Example 02
Check if 196 is a perfect square

Solution

(a) Do the prime factorization of number

(b) Check if all the factors are forming pair of two.

The above prime factorization is expressed as;

\mathtt{196\ =2^{2} \times 7^{2}}

Since all the factors have exponent 2, it means that the factors can be arranged in pair of two.

Hence, number 196 is a perfect square.

(c) To get the square root, remove the exponent and simply multiply the factor.

Square root of 196 ⟹ 2 x 7

Square root of 196 ⟹ 14

Hence, the square root of 196 is 14.

It means that if we multiply number 14 by itself, we will get 196.

Example 03
Check if number 32 is a perfect square.

Solution

(a) Do the prime factorization.

(b) Check if the factors are forming pair of two

The above factorization can be expressed as;

\mathtt{32\ =2^{2} \times 2^{2} \times 2}

Observe that all the factors doesn’t have exponent 2. This tells that the all the factors are not forming pair of two.

Hence, number 32 is not a perfect square.

Example 04
Check if the number 1089 is a perfect square.

Solution

(a) Do the prime factorization

(b) Check if the factors can be arranged in pair of two.

The above prime factorization can be represented as;

\mathtt{1089\ =3^{2} \times 11^{2}}

Observe that all the factors contain exponent 2, it means that all factors are forming pair of two.

Hence, the number 1089 is a perfect square.

(c) To get the square root, remove all the exponent and multiply the factors.

Square root of 1089 ⟹ 3 x 11

Square root of 1089 ⟹ 33

The square root of 1089 is 33. It means that if we multiply 33 by itself we will get number 33.

Example 05
Check if number 432 is a perfect square

Solution
(a) Do the prime factorization.

(b) Check if all the factors are forming pair of two.

The prime factorization can be represented as;

\mathtt{432\ =2^{2} \times 2^{2} \times 3^{3} \times 3}

Note that the factor 3 doesn’t have exponent of 2. It means that all the factors are not forming pair of two.

Hence, the number 432 is not a perfect square.