# Squares and Square Roots

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

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?

1. 720
2. 154
3. 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.

• 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. 122 = 144, 202 = 400
• Squares of odd numbers are always odd, e.g. 72 = 49, 212 = 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. 32 = 9, 122 = 144
• Examples of multiples of 3 exceeded by 1. 42 = 16 = (15+1), 132 = 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. 62 = 36, 82 = 64
• Examples of multiples of 4 exceeded by 1. 72 = 49 = (48+1), 92 = 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.
• 42 − 32 = 16 − 9 = 7 = 4 + 3
• The square of a natural number n is equal to the sum of the first n odd natural numbers. Example:
• 12 = 1 = sum of the first 1 odd natural number
• 22 = 1 + 3 = sum of the first 2 odd natural numbers
• 32 = 1 + 3 + 5 = sum of the first 3 odd natural numbers and so on.
• 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.

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

Now let us generalize our observations.

• Between 12 = 1 and 22 = 4, there are 2 non-square numbers (i.e., 2 x 1).
• Between 22 = 4 and 32 = 9, there are 4 non-square numbers (i.e., 2 x2).
• Between 32 = 9 and 42 = 16, there are 6 non-square numbers (i.e., 2 x 3).
• Between 52 = 25 and 62 = 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).

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