Class 8 Maths Squares and Cubes Notes

Class 8 Maths Squares and Cubes Notes

Squares and Cubes – Notes

🔢 1. What are Squares?

A square number is the product of a number multiplied by itself.

👉 Formula:
Square of a number = number × number = n²

✅ Examples:

  • 5² = 5 × 5 = 25
  • 12² = 144
  • 20² = 400

👉 These are called perfect squares.

🔍 2. Properties of Square Numbers

  • A square number always ends with: 0, 1, 4, 5, 6, or 9
  • It never ends with: 2, 3, 7, or 8
  • Number of zeros at the end is always even
  • Square of even number → even
  • Square of odd number → odd

📊 3. Finding Square Numbers

Method 1: Repeated Addition

👉 Example:
6² = 6 + 6 + 6 + 6 + 6 + 6 = 36

Method 2: Using Formula

👉 n² directly

✍️ 4. Square of Numbers (Shortcut Tricks)

(a + b)² Formula:

👉 (a + b)² = a² + 2ab + b²

Example:

(20 + 3)² =
= 20² + 2×20×3 + 3²
= 400 + 120 + 9 = 529

🔢 5. Square Roots

The square root of a number is the value which when multiplied by itself gives the number.

👉 √25 = 5
👉 √144 = 12

🧮 6. Finding Square Roots

Method 1: Prime Factorization

👉 Example: √36
= 2 × 2 × 3 × 3
= (2×3) = 6

Method 2: Division Method (Long Division)

  • Used for large numbers
  • Step-by-step grouping of digits

🔷 7. What are Cubes?

A cube number is a number multiplied by itself three times.

👉 Formula:
Cube = n × n × n = n³

✅ Examples:

  • 2³ = 8
  • 3³ = 27
  • 10³ = 1000

🔍 8. Properties of Cube Numbers

  • Cube of even number → even
  • Cube of odd number → odd
  • Cubes can end with any digit (0–9)
  • Cubes of negative numbers are negative

🧠 9. Cube Roots

The cube root of a number is the number which when multiplied 3 times gives the original number.

👉 ∛27 = 3
👉 ∛64 = 4

🧮 10. Finding Cube Roots

Method: Prime Factorization

👉 Example: ∛216
= 2 × 2 × 2 × 3 × 3 × 3
= (2 × 3) = 6

📐 11. Patterns in Squares and Cubes

Squares:

  • 1² = 1
  • 2² = 4
  • 3² = 9
  • Pattern increases by odd numbers

Cubes:

  • 1³ = 1
  • 2³ = 8
  • 3³ = 27

🌍 12. Uses in Real Life

  • Finding area of squares
  • Measuring volume of cubes
  • Used in engineering and construction
  • Helpful in calculations and problem solving

📝 13. Key Points to Remember

  • Square = number × number
  • Cube = number × number × number
  • Use prime factorization for roots
  • Learn squares up to 20 for quick solving
  • Practice shortcuts for faster calculation

Squares and Cubes – Practice Questions with Answers

🔢 Section A: Squares (1–15)

1. Find 12²
👉 144

2. Find 25²
👉 625

3. Find 30²
👉 900

4. Find 17²
👉 289

5. Find 9²
👉 81

6. Find 50²
👉 2500

7. Find 101²
👉 (100 + 1)² = 10000 + 200 + 1 = 10201

8. Find 99²
👉 (100 − 1)² = 10000 − 200 + 1 = 9801

9. Find 45²
👉 2025

10. Find square of 1.
👉 1

11. Find square of 0.
👉 0

12. Which is a perfect square: 121, 123, 125?
👉 121

13. Find square of 200
👉 40000

14. Find 19²
👉 361

15. Find square of 11
👉 121

🔍 Section B: Square Roots (16–25)

16. √64
👉 8

17. √121
👉 11

18. √225
👉 15

19. √400
👉 20

20. √144
👉 12

21. √169
👉 13

22. √256
👉 16

23. √81
👉 9

24. √1
👉 1

25. √10000
👉 100

🔷 Section C: Cubes (26–35)

26. Find 2³
👉 8

27. Find 5³
👉 125

28. Find 10³
👉 1000

29. Find 4³
👉 64

30. Find 7³
👉 343

31. Find 3³
👉 27

32. Find 6³
👉 216

33. Find 8³
👉 512

34. Find 9³
👉 729

35. Find 1³
👉 1

🧠 Section D: Cube Roots (36–45)

36. ∛8
👉 2

37. ∛27
👉 3

38. ∛64
👉 4

39. ∛125
👉 5

40. ∛216
👉 6

41. ∛343
👉 7

42. ∛512
👉 8

43. ∛729
👉 9

44. ∛1000
👉 10

45. ∛1
👉 1

📊 Section E: Mixed Practice (46–50)

46. Is 500 a perfect square?
👉 No

47. Is 729 a perfect cube?
👉 Yes

48. Find smallest number to multiply 50 to make it a perfect square
👉 50 = 2 × 5² → multiply by 2
👉 Answer: 2

49. Find square root of 625 using prime factorization
👉 625 = 5 × 5 × 5 × 5 → √625 = 25

50. Find cube root of 1728
👉 1728 = 12³ → ∛1728 = 12