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**

Follow the below steps;

**(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**

Follow the below steps;

**(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**

Follow the below steps;

**(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**

Follow the below steps;

(**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.**