Page 52 - Start Up Mathematics_8 (Non CCE)

3 Squares and Square Roots

We already know that the area of a square = a × a (where a is the length of a side) which means to find the
area of a square, we have to find the square of its side.
In this chapter, we will learn about some special operations that will help us to find squares and square roots
of numbers.

Perfect Square and Square Numbers

Square numbers or perfect squares are natural numbers whose exponent is 2, i.e., the number is raised to
the power 2. In other words, when a number is multiplied by itself, it is called its square. If x is any number,
2
then square of x = x × x = x .
2
2
2
For example, 3 = 3 × 3 = 9 4 = 4 × 4 = 16 11 = 11 × 11 = 121 and so on.
Table of squares of numbers from 1 to 30

Number Square Number Square Number Square Number Square
2
2
2
2
(x) (x ) (x) (x ) (x) (x ) (x) (x )
1 1 9 81 17 289 25 625
2 4 10 100 18 324 26 676

3 9 11 121 19 361 27 729

4 16 12 144 20 400 28 784

5 25 13 169 21 441 29 841

6 36 14 196 22 484 30 900

7 49 15 225 23 529
8 64 16 256 24 576

Prime Factorization Method to Check Perfect Squares

To check if a given natural number is a perfect square, it needs to be first broken down as a product of its
prime factors. The prime factors are then arranged in pairs. If no factor is left after pairing, the given natural
number is a perfect square, otherwise not.

Example 1: Is 625 a perfect square? 5 625
Solution: To find whether 625 is a perfect square or not, break it down as a 5 125
product of its prime factors. 5 25
\ 625 = 5 × 5 × 5 × 5 5 5

Since no factor is left unpaired, hence 625 is a perfect square. 1