**Squares and Square Roots:** Here, we shall be concentrating on the procedures to find the square roots of positive rational numbers.

## Table of Contents

## Squares

Look at the examples given below:

a × a = a²

2 × 2 = 4 = 2²

3 × 3 = 9 = 3²

Similarly, 4 × 4 = 16 = 4²

So, we conclude that, the square of a number is the product obtained by multiplying the number by itself only once.

Numbers, such as 1, 4, 9, 16, 25, 36 are called perfect squares.

**A given number is called a perfect square or a square number if it is the square of some natural number. These numbers are exact squares and do not Involve any decimals or fractions.**

To find out whether a given number is a perfect square or not, write the number as a product of its prime factors. If these factors exist in pairs, the number is a perfect square.

Let us take an example to find whether the given number is a perfect square or not.

**Example 1: Which of the following numbers are perfect squares?**

- 720
- 154
- 256

**Solution:**

**1. 720**

**Step 1:** 720 = **2×2** × **2×2** × **3×3** × 5

**Step 2:** In prime factors of 720, one factor 5 is left ungrouped.

**∴** 720 is not a perfect square.

**2. 154**

**Step 1:** 154 = 2 × 7 × 11

**Step 2:** No prime factor exists in pairs.

**∴** 154 is not a perfect square.

**3. 256**

**Step 1:** 256 = **2×2** × **2×2** × **2×2** × **2×2**

**Step 2:** Prime factors of 256 can be grouped into pairs and no factor is left out.

**⇒** 256 = (2 × 2 × 2 × 2)^{2} = (16)^{2}

**∴** 256 is a perfect square of 16.

## Facts About Perfect Squares

- A number ending with an odd number of zeros (one zero, three zeros and so on) is never a perfect square, e.g. 150, 25000, 350 are not perfect squares.
- Squares of even numbers are always even, e.g. 12
^{2}= 144, 20^{2}= 400 - Squares of odd numbers are always odd, e.g. 7
^{2}= 49, 21^{2}= 441 - The numbers ending with 2, 3, 7, 8 are not perfect squares, e.g. 32, 243, 37, 368 are not perfect squares.
- The square of a number other than 0 and 1, is either a multiple of 3 or exceeds the multiple of 3 by 1.
**Examples of multiples of 3.**3^{2}= 9, 12^{2}= 144**Examples of multiples of 3 exceeded by 1.**4^{2}= 16 = (15+1), 13^{2}= 169 = (168+1)

- The square of a number other than 0 and 1, is either a multiple of 4 or exceeds a multiple of 4 by 1.
**Examples of multiples of 4.**6^{2}= 36, 8^{2}= 64**Examples of multiples of 4 exceeded by 1.**7^{2}= 49 = (48+1), 9^{2}= 81 = (80+1)

- The difference between the squares of two consecutive natural numbers is equal to their sum. Let us take two consecutive natural numbers, 3 and 4.
- 4
^{2}− 3^{2}= 16 − 9 = 7 = 4 + 3

- 4
- The square of a natural number n is equal to the sum of the first n odd natural numbers. Example:
- 1
^{2}= 1 = sum of the first 1 odd natural number - 2
^{2}= 1 + 3 = sum of the first 2 odd natural numbers - 3
^{2}= 1 + 3 + 5 = sum of the first 3 odd natural numbers and so on.

- 1
- Squares of natural numbers composed of the only digit 1, follow a peculiar pattern.

We can also observe that the sum of the digits of every such number is a perfect square 1, 121, 12321, 1234321. See, how beautiful patterns of numbers are made below.

## Numbers between square numbers

Let us observe some interesting patterns between two consecutive square numbers in a tabular form:

Consecutive square numbers | Non-square numbers | Number of non-square numbers |
---|---|---|

1 and 4 | 2,3 | 2 |

4 and 9 | 5,6,7,8 | 4 |

9 and 16 | 10, 11, 12, 13, 14, 15 | 6 |

16 and 25 | 17, 18,……..,24 | 8 |

25 and 36 | 26, 27,……..,35 | 10 |

Now let us generalize our observations.

- Between 1
^{2}= 1 and 2^{2}= 4, there are 2 non-square numbers (i.e., 2 x 1). - Between 2
^{2}= 4 and 3^{2}= 9, there are 4 non-square numbers (i.e., 2 x2). - Between 3
^{2}= 9 and 4^{2}= 16, there are 6 non-square numbers (i.e., 2 x 3). - Between 5
^{2}= 25 and 6^{2}= 36, there are 10 non-square numbers (i.e., 2 x 5).

**There are 2n non-perfect square numbers between the square of the numbers, n and (n + 1).**

please also share ch 12…*needy*