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 calledperfect 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

📖 Next Steps

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