Class-8 Ganita prakash Worksheet Maths ch-1

Class-VIII Ganita prakash Worksheet   (2025-2026)                    Subject: Maths

                                                     

1. If a number ends in 0, 1, 4, 5, 6 or 9, is it always a square?

2. Write 5 numbers such that you can determine by looking at their units digit that they are not squares.

3. Which of the following numbers have the digit 6 in the unit's place?
   (i) 38² (ii) 34² (iii) 46² (iv) 56² (v) 74² (vi) 82²

4. If a number contains 3 zeros at the end, how many zeros will its square have at the end?

5. What can you say about the parity of a number and its square?

6. Using the sum of successive odd numbers pattern, find 36², given that 35² = 1225.

7. What is the nth odd number?
   a) 2n b) 2n-1 c) n +1 d) n+2

8. The 36th odd number is ____
   a) 71 b)81 c)101 d)91

9. Find how many numbers lie between two consecutive (m,m+1) perfect squares?
   a)2m b) 3m c) 4m d) m 

10. How many square numbers are there between 1 and 100? 

11. How many are between 101 and 200?

12.  What is the largest square less than 1000?

13.  Extend the pattern shown and draw the next term. 

14.  The area of a square is 49 sq. cm. What is the length of its side?

15. if y= x² then x is the square root of ______

16. What is the square root of 64?

17.   n2  = __________ a ) ± n b) 0 c) 2 d) -n,-m

18. find out if 576 or 327 is a perfect square? If it is a perfect square, find its square root? If not write the reason.

19. Is 324 a perfect square?

20. Is 156 a perfect square? 

21. Find whether 1156 and 2800 are perfect squares using prime factorisation

22. 1. Which of the following numbers are not perfect squares? (i) 2032 (ii) 2048 (iii) 1027 (iv) 1089 

23. 2. Which one among 642, 1082, 2922, 362 has last digit 4?

24.  3. Given 125² = 15625, what is the value of 126²? (i) 15625 + 126 (ii) 15625 + 26² (iv) 15625 + 251 (iii) 15625 + 253 (v) 15625 + 51² 

25. 4. Find the length of the side of a square whose area is 441 m². 

26. 5. Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10. 

27. 6. Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product. 

28. 7. How many numbers lie between the squares of the following numbers? (i) 16 and 17 (ii) 99 and 100

29. 8. In the following pattern, fill in the missing numbers 

1² + 2² + 2² = 3²

 2² + 3² + 6² = 7² 

3² + 4² + 12² = 13² 

4² + 5² + 20² = (___)²

 9² + 10² + (___)² = (___)² 

30. 9. How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.

31. How many cubes of side 1 cm will make a cube of side 3 cm?

32.  How many cubes of side 1 cm make a cube of side 2 cm?

33. Is 9 a cube? 

34.  estimate the number of unit cubes in a cube with an edge length of 4 units?

35. We know that 0, 1, 4, 5, 6, 9 are the only last digits possible for squares. What are the possible last digits of cubes?

36. Similar to squares, can you find the number of cubes with 1 digit, 2 digits, and 3 digits? What do you observe?

37. Can a cube end with exactly two zeroes (00)? Explain.

38.  The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which each of these can be expressed as the sum of two positive cubes.

39.  1 = 1 = 1³ 

3 + 5 = 8 = 2³

 7 + 9 + 11 = 27 = 3³

13 + 15 + 17 + 19 = 64 = 4³

21 + 23 + 25 + 27 + 29 = 125 = 5³ 

31 + 33 + 35 + 37 + 39 + 41 = 216 = 6³.

…………………. 

91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109.  what is this sum without doing the calculation?

40. check if 3375 is a perfect cube.

41. Is 500 a perfect cube?

42. Find the cube roots of these numbers: (i) 364  (ii)  3512  (iii)  3729  

43. Compute successive differences over levels for perfect cubes until all the differences at a level are the same. What do you notice? Perfect Cubes 1 8 27 64 125 216 ..

44. Find the cube roots of 27000 and 10648. 

45. 2. What number will you multiply by 1323 to make it a cube number? 

46. 3. State true or false. Explain your reasoning. 

(i) The cube of any odd number is even. 

(ii) There is no perfect cube that ends with 8. 

(iii) The cube of a 2-digit number may be a 3-digit number. 

(iv) The cube of a 2-digit number may have seven or more digits. 

(v) Cube numbers have an odd number of factors.

47. 4. You are told that 1331 is a perfect cube. Can you guess without factorisation what its cube root is? Similarly, guess the cube roots of 4913, 12167, and 32768

48. Square root is the inverse operation of _____________

49. A number obtained by multiplying a number by itself three times is called a ________.

50.  A number is a perfect cube if its prime factors can be split into _______ identical groups 

51. Which of the following is the greatest? Explain your reasoning.

 (i) 67³ – 66³ (ii) 43³ – 42³ (iii) 67² – 66² (iv) 43² – 42²

Square Pairs! CLASS 8 - Mathematics Subject Enrichment Activity-1

 Square Pairs!

CLASS 8 - Mathematics Subject Enrichment Activity-1
Topic:Number Patterns and Square Numbers 

Aim:

• To explore patterns formed by consecutive numbers whose sums are perfect squares.
  
  

• To develop logical reasoning and problem-solving skills through arranging consecutive numbers following given conditions.
  
  

Learning Objectives:

• Understand square numbers and their properties.
  
  

• Develop critical thinking to identify patterns in consecutive numbers.
  
  

• Apply systematic reasoning to solve arrangement puzzles.
  
  

Materials Required:

• Pencil and eraser
  
  

• Colored pencils or markers or colour papers
  
  

• Grid paper (optional for trials)
  
  

Procedure:

1. pairs of adjacent numbers add up to square numbers.
   For example:
   
   

• 3 + 6 = 9 (square)
  
  

• 6 + 10 = 16 (square)
  
  

• 10 + 15 = 25 (square)
  
  

• 15 + 1 = 16 (square)
  
  

2. Arranging the numbers from 1 to 17 (without repetition) so that each pair of adjacent numbers sums to a square number, recording the trials.

3. As an extension,  try to arrange numbers from 1 to 32 in a circular pattern under the same rules.
   

Observation:

• This puzzle is known and there is only one unique solution (up to reverse) for 1–17
  Verification (all adjacent sums are perfect squares):

• 16 + 9 = 25
  
  

• 9 + 7 = 16
  
  

• 7 + 2 = 9
  
  

• 2 + 14 = 16
  
  

• 14 + 11 = 25
  
  

• 11 + 5 = 16
  
  

• 5 + 4 = 9
  
  

• 4 + 12 = 16
  
  

• 12 + 13 = 25
  
  

• 13 + 3 = 16
  
  

• 3 + 6 = 9
  
  

• 6 + 10 = 16
  
  

• 10 + 15 = 25
  
  

• 15 + 1 = 16
  
  

• 1 + 8 = 9
  
  

• 8 + 17 = 25 

• This is an open mathematical challenge known as the "Square Sum Circle" or "Square Sum Ring." For 1 to 32, there does exist a valid circular solution.
  
  

Reflections:

• What strategies helped you solve the problem?
  
  

• Was there any part of the puzzle that seemed impossible? Why?
  
  

• How did you check whether your solution worked?
  
  

• What did you learn about square numbers and patterns?
  
  

Extension / Higher Order Thinking:

• Can you think of other patterns (like triangle numbers or prime sums) that could be used similarly?
  
  

• How would this problem change if you used negative numbers or fractions?

Reflections:

1. What strategies helped you solve the problem?

• I started by listing all square numbers up to 34 (since 17 + 16 = 33 is the maximum possible adjacent sum).

• Then, I listed all pairs of numbers from 1 to 17 (or 1 to 32) whose sums are perfect squares.

• Using these pairs, I created a graph where numbers were connected if they formed a valid square sum.

• I used logical trial and error and sometimes backtracked when I reached dead ends.

• Working in pairs helped — discussing logic and spotting patterns made the process faster.

2. Was there any part of the puzzle that seemed impossible? Why?

• Yes, at times it felt impossible to move forward because some numbers had very few valid square-sum partners.

• Especially near the end of the arrangement, I sometimes got stuck because the remaining numbers couldn’t be paired without breaking the square rule.

• Also, arranging numbers in a circle with no start or end made it more difficult than the linear version.

3. How did you check whether your solution worked?

• I added each pair of adjacent numbers to verify that the sum was a perfect square.

• In the circle arrangement, I made sure the last number and the first number also formed a square when added.

• I double-checked to ensure no number was repeated or missing in the final sequence.

4. What did you learn about square numbers and patterns?

• Square numbers follow a predictable pattern and appear frequently when combining smaller numbers.

• Some numbers, especially those in the middle of the range, have more square-sum partners than very small or very large numbers.

• This kind of puzzle shows how math and logic can come together to form fun, challenging patterns.

Extension / Higher Order Thinking:

5. Can you think of other patterns (like triangle numbers or prime sums) that could be used similarly?

• Yes! This puzzle could also be done with:

  • Triangle numbers (1, 3, 6, 10, 15, etc.) instead of squares.

  • Prime sums, where adjacent numbers must add to a prime number.

  • Fibonacci sums, where each pair adds to a Fibonacci number.

• Each of these would create a different type of logical puzzle and would require similar but adapted strategies.

6. How would this problem change if you used negative numbers or fractions?

• Using negative numbers would increase the number of possible pairs since the range of sums would widen.

• However, it could also make it harder to control the solution, and more combinations would need to be checked.

• If fractions were used, very few fractional sums would match perfect squares (which are always whole numbers), so the puzzle might become impossible or extremely limited.

• This shows that the set of allowed numbers plays a major role in how solvable a puzzle is.

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

CLASS 8 CHAPTER 2 POWER PLAY SOLUTIONS GANITA PRAKASH NCERT

Class 8 – Ganita Prakash

Chapter 2: Power Play – Full Answers with Explanations

🧠 2.1 Experiencing the Power Play

Thickness after 30 folds:
0.001 × 2³⁰ = 0.001 × 1,073,741,824 = 10.7 km

Thickness after 46 folds:
0.001 × 2⁴⁶ ≈ 703,687.4 km (Can reach the Moon!)

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📘 2.2 Exponential Notation and Operations

1. Express in exponential form:

• 6 × 6 × 6 × 6 = 6⁴
• y × y = y²
• b × b × b × b = b⁴
• 5 × 5 × 7 × 7 × 7 = 5² × 7³
• 2 × 2 × a × a = 2² × a²
• a × a × a × c × c × c × c × d = a³ × c⁴ × d

2. Prime factorisation:

• 648 = 2³ × 3⁴
• 405 = 3⁴ × 5
• 540 = 2² × 3³ × 5
• 3600 = 2⁴ × 3² × 5²

3. Numerical Values:

• 2 × 10³ = 2000
• 7² × 2³ = 392
• 3 × 4⁴ = 768
• (–3)² × (–5)² = 9 × 25 = 225
• 3² × 10⁴ = 9 × 10000 = 90,000
• (–2)⁵ × (–10)⁶ = –32,000,000

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💎 The Stones that Shine

Total diamonds = 3⁷ = 2187

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🧮 Properties of Exponents

• p⁴ × p⁶ = p¹⁰
• (4³)² = 4⁶ = 4096
• (2²)⁵ = 2¹⁰ = 1024
• (na)^b = n^a×b

Examples:

• 8⁶ = (2³)⁶ = 2¹⁸
• 7¹⁵ = (7⁵)³ or (7³)⁵
• 9¹⁴ = (3²)¹⁴ = 3²⁸

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🌸 Magical Pond

Day 30 = fully covered → Day 29 = half covered

Tripling pond after 4 days:
2⁴ × 3⁴ = 16 × 81 = 1296 lotuses

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👚 How Many Combinations

• Estu: 4 dresses × 3 caps = 12 combinations
• Roxie: 7 × 2 × 3 = 42 combinations
• 5-digit password: 10⁵ = 100,000
• 6-letter lock (A–Z): 26⁶ = 308,915,776

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✂️ 2.3 The Other Side of Powers

• n^a ÷ n^b = n^a–b
• n⁰ = 1 (n ≠ 0)
• n⁻ᵃ = 1/n^a

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🔬 2.4 Powers of 10 & Scientific Notation

Examples:

• 59,853 = 5.9853 × 10⁴
• 70,04,00,00,000 = 7.004 × 10¹⁰
• Earth–Sun = 1.496 × 10¹¹ m
• Moon = 3.844 × 10⁸ m → Need 3.844 × 10¹³ sheets of paper to reach

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📐 2.5 Did You Ever Wonder?

• Jaggery worth = 45 × ₹70 = ₹3,150
• Wheat worth = 50 × ₹50 = ₹2,500
• 1-rupee coins = 45,000 ÷ 4.8 ≈ 9,375 coins

Number of ladder steps to moon: 3,84,400 km ÷ 0.2 m = 1.922 × 10⁹ steps

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🧠 Figure It Out (Final Questions)

1. Bottles after 40 days:

5 × 40 = 200 bottles

2. 64³ – Different exponential forms:

• (2⁶)³ = 2¹⁸
• (8²)³ = 8⁶

3. True or False:

• Cube numbers are square numbers – Sometimes
• Fourth powers are square – Always
• Fifth power divisible by cube – Always
• Product of two cubes is a cube – Always
• q⁴⁶ is both 4th and 6th power (q is prime) – Yes

4. Simplify in exponential form:

• 10⁻² × 10⁻⁵ = 10⁻⁷
• 57 ÷ 54 = 5³ = 125
• 9⁻⁷ ÷ 9⁴ = 9⁻¹¹
• (13⁻²)⁻³ = 13⁶
• m⁵n¹²(mn)⁹ = m¹⁴n²¹

5. 122 = 144 ⇒ Then:

• (1.2)² = 1.44
• (0.12)² = 0.0144
• (0.012)² = 0.000144
• 120² = 14400

6. Scientific Calculations:

• World clothing = 8.2 × 10⁹ × 30 = 2.46 × 10¹¹
• Bee colonies = 100 million × 50,000 = 5 × 10⁹ × 5 × 10⁴ = 2.5 × 10¹⁴
• Bacteria in humans = 3.12 × 10²³
• Time spent eating = ~1.05 × 10⁸ seconds

7. Date 1 billion seconds ago from August 2025:

Approximately December 1993

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Prepared by: [KEYTOENJOYLEARNINGMATHS] – Based on NCERT Class 8 Chapter 2: Power Play