5.1 Introduction

You know that the area of a square = side × side (where ‘side’ means ’the length of a side’). Study the following table.

The numbers 1, 4, 9, 16, … are square numbers. These numbers are also called perfect squares.

5.2 Properties of Square Numbers

1. Square numbers end with 0, 1, 4, 5, 6, or 9 at unit’s place.

2. Square numbers have an even number of zeros at the end.

3. The square of an even number is even. The square of an odd number is odd.

4. The sum of first n odd numbers is n². 1 + 3 + 5 = 3² = 9 1 + 3 + 5 + 7 = 4² = 16

5.3 Patterns in Squares

Triangular Numbers: 1, 3, 6, 10, 15, 21, … are triangular numbers.

Square Numbers: 1, 4, 9, 16, 25, 36, … are square numbers.

Relationship: Sum of two consecutive triangular numbers = square number 3 + 6 = 9 = 3² 6 + 10 = 16 = 4²

5.4 Finding Square Root

Square Root: The square root of a number is that number which when multiplied by itself gives the original number.

Methods to find square root:

1. Repeated Subtraction Method
2. Prime Factorization Method
3. Division Method

Method 1: Repeated Subtraction

Find √36: 36 - 1 = 35 (1st odd number) 35 - 3 = 32 (2nd odd number) 32 - 5 = 27 (3rd odd number) 27 - 7 = 20 (4th odd number) 20 - 9 = 11 (5th odd number) 11 - 11 = 0 (6th odd number)

We subtracted 6 times, so √36 = 6

Method 2: Prime Factorization

Find √324: 324 = 2 × 2 × 3 × 3 × 3 × 3 324 = 2² × 3² × 3² 324 = (2 × 3 × 3)² √324 = 2 × 3 × 3 = 18

5.5 Square Roots of Decimals

Example: Find √2.56

Step 1: Remove decimal → 256 Step 2: Find √256 = 16 Step 3: Count decimal places in original number (2) Step 4: Place decimal point → √2.56 = 1.6

5.6 Estimating Square Roots

Example: Estimate √300

We know: 17² = 289 and 18² = 324 Since 300 is between 289 and 324, √300 is between 17 and 18

5.7 Word Problems

Example 1: Find the smallest number by which 180 must be multiplied to get a perfect square.

Solution: 180 = 2² × 3² × 5 To make it a perfect square, multiply by 5 180 × 5 = 900 = 30²

Example 2: A square field has area 2025 m². Find its side length.

Solution: Side = √2025 = 45 m

Key Points to Remember:

• Perfect squares end with 0, 1, 4, 5, 6, or 9
• Square of even number is even, square of odd number is odd
• Three methods to find square roots: repeated subtraction, prime factorization, division
• √(a × b) = √a × √b
• √(a ÷ b) = √a ÷ √b

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