Lesson Notes: Squares and Square Roots

1. Squares of Numbers

• Square of a number: When a number is multiplied by itself. Example: 5² = 5 × 5 = 25
• Perfect Square: A number that can be expressed as the square of an integer. Example: 1, 4, 9, 16, 25, …
• Properties of Squares:
  • A number ending in 2, 3, 7, or 8 cannot be a perfect square.
  • Perfect squares of even numbers are even, and those of odd numbers are odd.
  • The number of zeros at the end of a perfect square is always even.

2. Square Roots

• Square Root: The inverse operation of squaring a number. Example: √36 = 6 since 6² = 36.
• Perfect Square Root: If a number’s square root is a whole number, the number is a perfect square. Example: √49 = 7.

3. Properties of Square Roots

• Non-perfect squares have irrational square roots (e.g., √2, √3).
• The square root of an even perfect square is even, and the square root of an odd perfect square is odd.

4. Finding the Square of a Number

• Formula for (a + b)²: (a + b)² = a² + 2ab + b²
• Formula for (a – b)²: (a - b)² = a² - 2ab + b²

5. Finding Square Roots Using Different Methods

(i) Prime Factorization Method

Express the number as a product of prime factors, and pair similar factors.

Example: 144 = 2 × 2 × 2 × 2 × 3 × 3
Taking one factor from each pair: 2 × 2 × 3 = 12
So, √144 = 12.

(ii) Long Division Method

Used to find the square roots of large numbers. Follow step-by-step division to find square roots accurately.

6. Pythagorean Triplets

A set of three numbers (a, b, c) that satisfy the relation a² + b² = c².

Formula to generate Pythagorean triplets: 2m, m² - 1, m² + 1 for any positive integer m > 1.

Example: For m = 2, the triplet is (4, 3, 5).

7. Estimating Square Roots

For numbers that are not perfect squares, we estimate between two nearby perfect squares.

Example: √50 lies between √49 = 7 and √64 = 8.

8. Applications of Squares and Square Roots

• Area of a square: Area = side²
• Diagonal of a square: d = √2 × side
• Used in solving quadratic equations and applying Pythagoras theorem.

9. Practice Questions

1. Find the square of 23.
2. Check if 225 is a perfect square.
3. Find √400 using the prime factorization method.
4. Estimate √90 between two perfect squares.
5. Identify a Pythagorean triplet starting with 6.