### **Chapter 2: Power Play**
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#### **Basic Concepts**
- \( n^a = n \times n \times n \times ... \times n \) (a times)
- \( n^{-a} = \frac{1}{n^a} \)
- \( n^a \times n^b = n^{a+b} \)
- \( (n^a)^b = n^{a \times b} \)
- \( n^a \times m^a = (n \times m)^a \)
- \( n^0 = 1 \) (n ≠ 0)
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#### **Worksheet Answers**
1. \( n \times n = n^2 \) (read as 'n squared' or 'n to the power 2')
2. \( n^a \) denotes n multiplied by itself **a** times.
3. \( a \times a \times a \times b \times b = a^3b^2 \) → (d)
4. Thickness after 10 folds: \( 2^{10}v \) → (c)
5. \( 32400 = 2^4 \times 3^4 \times 5^2 \)
6. \( (-1)^5 = -1 \) (negative), \( (-1)^{56} = 1 \) (positive)
7. \( (-2)^4 = 16 \) ✅
8. (i) \( 6^4 \), (ii) \( b^4y \), (iii) \( 5^2 \times 7^2 \), (iv) \( 2^2 \times a^2 \), (v) \( a^3 \times c^3 \times d \)
9. (i) \( 648 = 2^3 \times 3^4 \), (ii) \( 405 = 3^4 \times 5 \), (iii) \( 540 = 2^2 \times 3^3 \times 5 \), (iv) \( 3600 = 2^4 \times 3^2 \times 5^2 \)
10. (i) 2000, (ii) 392, (iii) 768, (iv) 225, (v) 90000, (vi) \( -32 \times 10^6 = -32,000,000 \)
11. \( 3^7 = 3^2 \times 3^5 \)
12. \( p^4 \times p^6 = p^{10} \)
13. (i) \( 2^9 = 512 \), (ii) \( 5^7 = 78125 \), (iii) \( 4^6 = 4096 \)
14. \( 7^4 = 2401 \)
15. \( 2^{10} = (2^5)^2 = 32^2 \)
16. (i) \( 8^6 = (2^3)^6 = 2^{18} \), (ii) \( 7^{15} = (7^5)^3 \), (iii) \( 9^{14} = (9^7)^2 \), (iv) \( 5^8 = (5^4)^2 \)
17. (i) Fully covered: \( 2^{30} \), (ii) Half covered: \( 2^{29} \)
18. \( 2^4 \times 3^4 = 16 \times 81 = 1296 \)
19. \( \frac{10^4}{5} = 2000 = 2 \times 10^3 \)
20. Estu: \( 4 \times 3 = 12 \), Roxie: \( 7 \times 2 \times 3 = 42 \)
21. \( 10^5 = 100,000 \) passwords
22. \( 2^{100} + 2^{25} \) (cannot be simplified further)
23. \( 2^0 = 1 \)
24. (i) \( \frac{1}{2^4} \), (ii) \( \frac{1}{10^5} \), (iii) \( \frac{1}{(-7)^2} \), (iv) \( \frac{1}{(-5)^3} \), (v) \( \frac{1}{10^{100}} \)
25. (i) \( 2^3 \), (ii) \( 3^3 \), (iii) \( 2^4 \times \frac{1}{(-4)^2} = \frac{16}{16} = 1 \), (iv) \( 8^{p+q} \)
26. \( 4^7 = 16,384 \), \( 4^5 = 1024 \), \( 16,384 ÷ 1024 = 16 \) → ✅
27. \( 4^2 ÷ 4^{-2} = 16 ÷ \frac{1}{16} = 256 \)
28. Use power line for 7 (values: 1, 7, 49, 343, 2401, 16807, 117649, 823543)
29. Arrange powers of 4: \( 4^1 = 4 \), \( 4^2 = 16 \), \( 4^3 = 64 \), \( 4^4 = 256 \), \( 4^5 = 1024 \), \( 4^6 = 4096 \), \( 4^7 = 16384 \), \( 4^8 = 65536 \)
30. (i) 172, (ii) 5642, (iii) 6374, (iv) 47561 (no specific instruction; likely standard form)
31. \( 561.903 = 5.61903 \times 10^2 \)
32. (i) \( 3.0 \times 10^{20} \), (ii) \( 1.0 \times 10^{11} \), (iii) \( 5.976 \times 10^{24} \), (iv) \( 5.9 \times 10^3 \), (v) \( 2.08 \times 10^4 \), (vi) \( 8.0 \times 10^6 \)
33. Smallest distance: Sun-Earth (\( 1.496 \times 10^{11} \))
34. Scientific notation: \( x \times 10^y \), \( 1 ≤ x < 10 \), y integer
35. Earth at \( 1.496 \times 10^{11} \), Saturn at \( 1.4335 \times 10^{12} \)
36. (i) \( 5.9853 \times 10^4 \), (ii) \( 3.43 \times 10^6 \), (iii) \( 6.595 \times 10^4 \), (iv) \( 7.004 \times 10^9 \)
37. Assume 80 years, walking speed 5 km/h → approx. 35,040 times
38. (i) \( 10^{22} \) seconds (approx.), (ii) \( 6.9 \times 10^{19} \) seconds (approx.)
39. Units digit of \( 2^{224} + 4^{32} = 6 + 6 = 12 \) → **2**
40. \( 5^{40} \) bottles
41. (i) \( 64^3 = (8^2)^3 = 8^6 \), (ii) \( 192^8 = (64 \times 3)^8 = 2^{48} \times 3^8 \), (iii) \( 32^{-5} = (2^5)^{-5} = 2^{-25} \)
42. (i) Sometimes (e.g., 64), (ii) Always (\( (a^2)^2 = a^4 \)), (iii) Always (\( a^5 ÷ a^3 = a^2 \)), (iv) Always (\( a^3 \times b^3 = (ab)^3 \))
43. (v) Sometimes (if q is a perfect 12th power)
44. (i) \( 10^{-7} \), (ii) \( 5^{11} \), (iii) \( 9^{-3} \), (iv) \( 13^6 \), (v) \( m^4 n^{11} \)
45. (i) 1.44, (ii) 0.0144, (iii) 0.000144, (iv) 14400
46. Same: (ii) \( 6^4 \times 3^2 = 1296 \times 9 = 11664 \), (iv) \( 18^2 \times 6^2 = 324 \times 36 = 11664 \)
47. (i) \( 3^4 = 81 > 4^2 = 16 \), (ii) \( 2^8 = 256 > 8^2 = 64 \), (iii) \( 2^{100} > 100^2 \)
48. \( 8.5 \times 10^9 \) packets → 10-digit code (\( 10^{10} > 8.5 \times 10^9 \))
49. Yes, e.g., 64, 729 → \( n^6 \)
50. \( 36^5 \) (26 letters + 10 digits)
51. \( 10^9 + 10^9 = 2 \times 10^9 \) → (v)
52. (i) \( 8 \times 10^9 \times 30 = 2.4 \times 10^{11} \), (ii) \( 10^8 \times 5 \times 10^4 = 5 \times 10^{12} \), (iii) \( 38 \times 10^{12} \times 8 \times 10^9 = 3.04 \times 10^{23} \), (iv) Assume 80 yrs × 365 × 24 × 3600 ≈ \( 2.52 \times 10^9 \) sec
53. 1 billion seconds ≈ 31.7 years ago
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### **Assertion and Reasoning**
1. (A)
2. (C) – R is false (base multiplied exponent times)
3. (A)
4. (A)
5. (D) – A is false (exponential grows faster)
6. (A)
7. (A)
8. (A)
9. (D) – A is false (even³ = even)
10. (A)
11. (A)
12. (A)
13. (A)
14. (C) – A true, R false (exponential faster long-term)
15. (D) – A true, R false (right shift → positive exponent)
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### **Case Study 1: Paper Folding**
1. (b) 0.002 cm
2. (a) 1 cm
3. (c) height of a person
4. (a) Eiffel Tower (330 m)
5. (b) airplane flying height
6. (b) 10 m
7. (d) 8 times
8. (b) thickness doubles every fold
9. (b) the Moon
10. (c) geometric progression
11. \( 0.001 \times 2^{10} = 1.024 \) cm
12. \( 0.001 \times 2^{46} \) cm ≈ 70,368 km → \( 7.0368 \times 10^4 \) km
13. Exponential growth is much faster than linear.
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### **Case Study 2: King’s Diamonds**
1. (c) 9
2. (c) 27
3. (c) 81
4. (b) 729
5. (d) \( 3^7 \)
6. (c) 27
7. (b) powers & exponents
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### **Case Study 3: Magical Lotus Pond**
1. (b) 29th day
2. (a) \( 2^{29} \)
3. (b) \( 2 \times 2^{30} \) (incorrect; should be \( \frac{1}{2} \times 2^{30} = 2^{29} \))
4. (c) \( 2^4 = 16 \)
5. (c) \( 16 \times 3^4 \)
6. (b) 1296
7. (b) same
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### **Case Study 4: Dresses, Caps & Shoes**
1. (b) 12
2. (b) \( 4 \times 3 \)
3. (c) 42
4. (b) \( 7 \times 2 \times 3 \)
5. (a) \( 5 \times 4 = 20 \)
6. (b) multiplication principle
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### **Case Study 5: Password**
1. (c) 100
2. (b) 1000
3. (c) 100,000
4. (b) \( 26^6 \)
5. (d) all
6. \( 10^5 = 100,000 \)
7. \( 26^5 \)
8. 6-letter (A-Z) is more secure (\( 26^6 > 10^5 \))
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### **Case Study 6: TulΔbhΔra Donation**
1. (c) ₹3150
2. (b) ₹2500
3. (c) 9000 coins
4. (b) Bhakti
5. (b) 10000 coins
6. ₹3150 + ₹2500 = ₹5650
7. \( \frac{45000}{4} = 11250 \) coins
8. \( 1.125 \times 10^4 \)
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**Prepared by Chithra Dhananjayan TGT Maths**
**Class VIII – Ganita Prakash (2025–2026)**
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