Chapter 1: A Square and a Cube of Class 8 – NCERT Ganita Prakash.

Chapter 1: A Square and a Cube of Class 8 – NCERT Ganita Prakash.

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📘 Chapter 1: A Square and a Cube – Full Answer Key with Explanations

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🔹 1.1 Seeing Squares All Around

Q: What is a square number?

• A number that is the product of a number multiplied by itself.
  E.g., 1² = 1, 2² = 4, 3² = 9, etc.

Figure it Out (Page 4)
Q: Numbers between 1 and 100 which are perfect squares:
→ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
(Total: 10 perfect squares)

Q: How many rectangles are in a 10×10 square grid?

• Total rectangles = n(n + 1)/2 × n(n + 1)/2

• For 10×10:
  = 10×11/2 × 10×11/2 = 55 × 55 = 3025

Q: How many of them are squares?

• Total squares in n×n grid = 1² + 2² + ... + 10² = 385

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🔹 1.2 Properties of Perfect Squares

Q: Unit digits of perfect squares:
Can only end with 0, 1, 4, 5, 6, 9
(E.g., 16 → 6; 25 → 5)

Figure it Out (Page 6)
Which of the following numbers are NOT perfect squares?

• 252 ⇒ Not a perfect square (ends in 2)

• 397 ⇒ Not (ends in 7)

• 444 ⇒ Not (ends in 4, but 21² = 441; 22² = 484 → so 444 not between)

• 405 ⇒ Not (20² = 400, 21² = 441 → not square)

• 529 ⇒ Yes (23² = 529)

• 729 ⇒ Yes (27² = 729)

• 841 ⇒ Yes (29² = 841)

✅ Perfect Squares: 529, 729, 841

Q: Are all even numbers perfect squares?
→ No. Example: 2, 6, 10 – none are squares.

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🔹 1.3 Playing with Patterns

Q: Sum of consecutive odd numbers gives square numbers
Example:
1 = 1²
1 + 3 = 4 = 2²
1 + 3 + 5 = 9 = 3²
1 + 3 + 5 + 7 = 16 = 4²
... and so on.

Figure it Out (Page 8)
Q: Find 9² = ?
→ Sum of 9 consecutive odd numbers:
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81

Q: Visual pattern: How many matchsticks used?

• For 1 square = 4 sticks

• For 2 squares (joined) = 7

• General rule:
  Matchsticks = 3n + 1 (for n squares)

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🔹 1.4 Finding Square Roots

Q: What is a square root?
→ If x² = y, then x is the square root of y.
E.g., √25 = 5

Methods:

• Prime factorisation

• Long division

Figure it Out (Page 10)
Q: Find square roots using prime factorisation:

• √144 = √(2⁴ × 3²) = 2² × 3 = 12

• √169 = √(13²) = 13

• √256 = √(2⁸) = 2⁴ = 16

• √196 = √(2² × 7²) = 2 × 7 = 14

Q: Is √17 a rational number?
→ No, 17 is not a perfect square. So, √17 is irrational.

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🔹 1.5 Making Cubes

Q: Cube of a number = number × number × number
E.g., 2³ = 8, 3³ = 27

Q: Is 16 a cube number?
→ No (2³ = 8, 3³ = 27 → 16 not between any)

Q: Is 64 a cube number?
→ Yes, 4³ = 64

Q: First 10 cube numbers:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Figure it Out (Page 13)
Q: Which are cube numbers?

• 8 (Yes)

• 64 (Yes)

• 216 (Yes)

• 343 (Yes)

• 1000 (Yes)

• 729 (Yes)

• 512 (Yes)

• 100 (No)

• 90 (No)

• 121 (No)

✅ Cube numbers: 8, 64, 216, 343, 512, 729, 1000

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🔹 1.6 Playing with Cubes

Q: Is 2³ + 3³ = 5³?
→ No. 8 + 27 = 35 ≠ 125

Q: Can sum of cubes of two numbers be a cube?
→ Only in rare special cases. (e.g., 1³ + 2³ = 9, not cube)

Figure it Out (Page 14)
Q: Check if the following are equal:
(i) 2³ + 3³ = 8 + 27 = 35
(ii) 4³ + 5³ = 64 + 125 = 189
→ Neither are cube numbers.

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🔹 1.7 Finding Cube Roots

Q: Cube root (∛) is the number that when cubed gives the number.
Example: ∛27 = 3 because 3³ = 27

Using prime factorisation:

• ∛512 = ∛(2⁹) = 2³ = 8

• ∛343 = ∛(7³) = 7

• ∛216 = ∛(2³ × 3³) = 2 × 3 = 6

Figure it Out (Page 15)
Find cube roots:

• ∛64 = 4

• ∛125 = 5

• ∛1000 = 10

• ∛729 = 9

• ∛27 = 3

• ∛100 = not a perfect cube

• ∛343 = 7

• ∛1 = 1

✅ Valid cube roots: 4, 5, 10, 9, 3, 7, 1

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🔹 1.8 Numbers and Their Last Digits

Q: Unit digit of square and cube numbers:

• Square numbers end in: 0, 1, 4, 5, 6, 9

• Cube numbers: any digit (0–9) possible

Figure it Out (Page 16)
Find unit digit of:

• 17² = 289 → 9

• 21² = 441 → 1

• 13³ = 2197 → 7

• 14³ = 2744 → 4

• 19³ = 6859 → 9

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🔹 1.9 A Puzzle

Q: Number x such that:

• Square of x ends in 25

• Cube of x ends in 125

→ Try:
25² = 625 (ends in 25)
25³ = 15625 (ends in 625, not 125)
Try:
5² = 25
5³ = 125 ✅

✅ Answer: 5

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🔹 Final Figure It Out (Page 17)

1. Sum of first 6 odd numbers = ?
   1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²

2. Is 196 a perfect square?
   Yes → 14² = 196 ✅

3. Is 256 a perfect cube?
   No → ∛256 is irrational

4. Is √289 rational?
   Yes → √289 = 17

5. Cube root of 1728?
   → ∛1728 = 12 ✅

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✅ Summary Table

Concept	Example	Result
Square of 13	13²	169
Cube of 9	9³	729
√121	–	11
∛343	–	7
Perfect Squares ≤ 100	–	10
Total Squares in 10×10 Grid	–	385
Total Rectangles in 10×10 Grid	–	3025