2. Power Play

After 46 folds, the thickness is about 7,03,687 km. This is more than the distance to the Moon (3,84,400 km).

After two folds the thickness increases 4 times (2 × 2). For example, 0.016 cm × 4 = 0.064 cm (fold 4 to fold 6).

After three folds it increases 2 × 2 × 2 = 8 times. For example, 0.016 cm × 8 = 0.128 cm (fold 4 to fold 7). So it is true.

The thickness doubles 10 times. So it is v × 2 × 2 × … (10 times).

Answer: (v) 2¹⁰v

(−1)⁵ = −1. It is negative because 5 is odd.

(−1)⁵⁶ = 1. It is positive because 56 is even.

(−2) × (−2) × (−2) × (−2) = 4 × 4 = 16. Yes, it is true.

0² = 0 and 0⁵ = 0. For any counting number n, 0ⁿ = 0.

(i) 6 × 6 × 6 × 6 = 6⁴

(ii) y × y = y²

(iii) b × b × b × b = b⁴

(iv) 5 × 5 × 7 × 7 × 7 = 5² × 7³

(v) 2 × 2 × a × a = 2²a²

(vi) a × a × a × c × c × c × c × d = a³c⁴d

(i) 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2³ × 3⁴

(ii) 405 = 3 × 3 × 3 × 3 × 5 = 3⁴ × 5

(iii) 540 = 2 × 2 × 3 × 3 × 3 × 5 = 2² × 3³ × 5

(iv) 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 = 2⁴ × 3² × 5²

(i) 2 × 10³ = 2 × 1000 = 2000

(ii) 7² × 2³ = 49 × 8 = 392

(iii) 3 × 4⁴ = 3 × 256 = 768

(iv) (−3)² × (−5)² = 9 × 25 = 225

(v) 3² × 10⁴ = 9 × 10000 = 90000

(vi) (−2)⁵ × (−10)⁶ = −32 × 1000000 = −3,20,00,000

Daughters = 3, baskets = 3 × 3 = 9, keys = 27, rooms = 3⁴ = 81

Tables = 3⁵ = 243, necklaces = 3⁶ = 729.

Diamonds = 3⁷ = 3⁴ × 3³ = 81 × 27 = 2187

3⁷ = 3 × 3 × 3 × 3 × 3 × 3 × 3 = (3 × 3) × (3 × 3 × 3 × 3 × 3) = 3² × 3⁵.

We only regroup the seven 3s into a group of 2 and a group of 5.

(i) 2⁹ = 2⁴ × 2⁵ = 16 × 32 = 512

(ii) 5⁷ = 5³ × 5⁴ = 125 × 625 = 78125

(iii) 4⁶ = 4³ × 4³ = 64 × 64 = 4096

(i) 8⁶ = (8²)³ = (8³)²

(ii) 7¹⁵ = (7³)⁵ = (7⁵)³

(iii) 9¹⁴ = (9²)⁷ = (9⁷)²

(iv) 5⁸ = (5²)⁴ = (5⁴)²

The pond starts with 1 lotus and the number doubles every day.

(i) Fully covered (30th day): 2³⁰

(ii) Half covered (29th day): 2²⁹

2⁵ × 5⁵ = (2 × 5)⁵ = 10⁵ = 1,00,000

10⁴ ÷ 5⁴ = (10 ÷ 5)⁴ = 2⁴ = 16

For each dress there are 2 hats, and for each of these there are 3 pairs of shoes.

Number of ways = 7 × 2 × 3 = 42

Each slot has 26 choices.

Passwords = 26 × 26 × 26 × 26 × 26 × 26 = 26⁶ = 30,89,15,776

2¹⁰⁰ ÷ 2²⁵ = 2^{100 − 25} = 2⁷⁵

If n = 0, then n^b = 0. We cannot divide by 0. So n cannot be 0.

Yes. The rules n^a × n^b = n^{a + b}, (n^a)^b = n^ab and n^a ÷ n^b = n^{a − b} hold for all integers a and b (n ≠ 0).

(i) 2⁻⁴ = 12⁴ = 116

(ii) 10⁻⁵ = 110⁵ = 11,00,000

(iii) (−7)⁻² = 1(−7)² = 149

(iv) (−5)⁻³ = 1(−5)³ = −1125

(v) 10⁻¹⁰⁰ = 110¹⁰⁰

(i) 2⁻⁴ × 2⁷ = 2^{−4 + 7} = 2³

(ii) 3² × 3⁻⁵ × 3⁶ = 3^{2 − 5 + 6} = 3³

(iii) p³ × p⁻¹⁰ = p⁻⁷

(iv) 2⁴ × (−4)⁻² = 16 × 116 = 1 = 2⁰

(v) 8^p × 8^q = 8^{p + q}

4² ÷ 4⁻² = 4^{2 − (−2)} = 4⁴ = 256 times

2401 × 49 = 7⁴ × 7² = 7⁶ = 1,17,649

49³ = (7²)³ = 7⁶ = 1,17,649

343 × 2401 = 7³ × 7⁴ = 7⁷ = 8,23,543

16,80749 = 7⁵ ÷ 7² = 7³ = 343

7343 = 7^{1 − 3} = 7⁻² = 149

16,8078,23,543 = 7^{5 − 7} = 7⁻² = 149

1,17,649 × 1343 = 7⁶ × 7⁻³ = 7³ = 343

1343 × 1343 = 7⁻³ × 7⁻³ = 7⁻⁶

(i) 172 = (1 × 10²) + (7 × 10¹) + (2 × 10⁰)

(ii) 5642 = (5 × 10³) + (6 × 10²) + (4 × 10¹) + (2 × 10⁰)

(iii) 6374 = (6 × 10³) + (3 × 10²) + (7 × 10¹) + (4 × 10⁰)

(i) Distance of the Sun from the centre of the Milky Way = 3 × 10²⁰ m

(ii) Number of stars in our galaxy = 1 × 10¹¹

(iii) Mass of the Earth = 5.976 × 10²⁴ kg

Sun–Saturn = 1.4335 × 10¹² m, Saturn–Uranus = 1.439 × 10¹² m, Sun–Earth = 1.496 × 10¹¹ m.

The Sun–Earth distance has the smallest power of 10. So it is the smallest.

1.496 × 10¹¹ ÷ 1.4335 × 10¹² ≈ 0.1. So the Earth is marked at about one-tenth of the way from the Sun to Saturn.

(i) 59,853 = 5.9853 × 10⁴

(ii) 65,950 = 6.595 × 10⁴

(iii) 34,30,000 = 3.43 × 10⁶

(iv) 70,04,00,00,000 = 7.004 × 10¹⁰

Guess first. Then use assumptions.

Let Roxie’s weight = 45 kg = 45,000 g. Let a 1-rupee coin weigh about 4 g.

Number of coins = 45,000 ÷ 4 ≈ 11,250. So the number is in ten-thousands (about 1 × 10⁴).

Let a 5-rupee coin weigh about 6 g. Number of coins = 45,000 ÷ 6 = 7,500. Money = 7,500 × 5 = ₹37,500.

Let a 10-rupee note weigh about 1 g. Number of notes = 45,000. Money = 45,000 × 10 = ₹4,50,000.

Let one meal need about 250 g of grain. Grain = 45 kg = 45,000 g.

Meals = 45,000 ÷ 250 = 180. So about 180 people can get one meal.

Let a person walk about 25 km in a day.

Days = 400 ÷ 25 = 16 days. So they started about two weeks ago.

Let the speed be 5 km per hour and the lifetime be 70 years.

Distance = 5 × 24 × 365 × 70 = 30,66,000 km.

Number of times = 30,66,000 ÷ 40,000 ≈ 76 times

Linear growth: saving ₹10 every day; climbing stairs one step at a time; a car moving at a fixed speed.

Exponential growth: bacteria doubling every hour; a message forwarded to 2 new people by each person; paper folding.

Starlings = 1.3 × 10⁹. Mosquitoes = 110 trillion = 1.1 × 10¹⁴

People per elephant = (8 × 10⁹) ÷ (4 × 10⁵) = 2 × 10⁴ = 20,000. So yes, we can say this.

Roxie: 4840 × 24 = 1,16,160 hours

Estu: 4070 days ≈ 11 years 1 month. Counting 4070 days back from 9 October 2026 gives 18 August 2015.

A million seconds = 10⁶ ÷ (60 × 60 × 24) ≈ 11.6 days

(i) A weekend of 2 days ≈ 1.7 × 10⁵ seconds; a train journey from Mumbai to Delhi of about 1 day ≈ 8.6 × 10⁴ seconds.

(ii) Diwali vacation of 2 weeks ≈ 1.2 × 10⁶ seconds; one month ≈ 2.6 × 10⁶ seconds.

One year ≈ 3.15 × 10⁷ seconds.

15 million years ≈ 1.5 × 10⁷ × 3.15 × 10⁷ ≈ 4.7 × 10¹⁴ seconds

47 crore years ≈ 4.7 × 10⁸ × 3.15 × 10⁷ ≈ 1.5 × 10¹⁶ seconds

The first part is a number prefix: bi = 2, tri = 3, quadri = 4, quinti = 5, and so on.

It tells how many times 1000 is multiplied with the first 1000. For example, billion = 1000 × 1000² = 10⁹, trillion = 1000 × 1000³ = 10¹².

4³² = (2²)³² = 2⁶⁴. So 2²²⁴ ÷ 2⁶⁴ = 2¹⁶⁰.

Units digits of powers of 2 repeat as 2, 4, 8, 6. 160 is divisible by 4.

So the units digit of 2¹⁶⁰ is 6

One container comes each day. In 40 days, 40 containers come.

Bottles = 40 × 5 = 200 = 2 × 10².

(i) 64³ = 2¹⁸: 2¹⁰ × 2⁸; 4⁵ × 4⁴; 8³ × 8³

(ii) 192⁸ = (2⁶ × 3)⁸: 2⁴⁸ × 3⁸; 2⁴⁰ × 6⁸; 64⁸ × 3⁸

(iii) 32⁻⁵ = 2⁻²⁵: 2⁻¹⁰ × 2⁻¹⁵; 2⁻⁵ × 2⁻²⁰; 4⁻¹² × 2⁻¹

(i) 10⁻² × 10⁻⁵ = 10⁻⁷

(ii) 5⁷ ÷ 5⁴ = 5³

(iii) 9⁻⁷ ÷ 9⁴ = 9^{−7 − 4} = 9⁻¹¹

(iv) (13⁻²)⁻³ = 13⁶

(v) m⁵n¹²(mn)⁹ = m^{5 + 9}n^{12 + 9} = m¹⁴n²¹

(i) (1.2)² = 1.44

(ii) (0.12)² = 0.0144

(iii) (0.012)² = 0.000144

(iv) 120² = 14400

6⁴ × 3² = 2⁴ × 3⁴ × 3² = 2⁴ × 3⁶.

18² × 6² = (2 × 3²)² × (2 × 3)² = 2⁴ × 3⁶.

So 2⁴ × 3⁶, 6⁴ × 3² and 18² × 6² are the same (= 11664).

(i) 4³ = 64 and 3⁴ = 81. So 3⁴ is greater.

(ii) 2⁸ = 256 and 8² = 64. So 2⁸ is greater.

(iii) 100² = 10⁴, but 2¹⁰⁰ is about 1.27 × 10³⁰. So 2¹⁰⁰ is greater.

With n digits, 10ⁿ codes are possible.

10⁹ = 1 billion is less than 8.5 billion. 10¹⁰ = 10 billion is more than 8.5 billion.

So the code should have 10 digits

Yes. Such numbers are sixth powers, n⁶ = (n³)² = (n²)³.

Examples: 1⁶ = 1, 2⁶ = 64, 3⁶ = 729, 4⁶ = 4096.

Each place can have 26 letters or 10 digits, that is 36 choices.

Codes = 36 × 36 × 36 × 36 × 36 = 36⁵ = 6,04,66,176

Total = 10⁹ + 10⁹ = 2 × 10⁹.

So (v) and (vi) are correct. Both mean the same number.

10⁹ seconds ÷ 86,400 ≈ 11,574 days ≈ 31.7 years.

Counting 11,574 days back from 9 October 2026 gives about 31 January 1995.

Roxie’s number = 4 × 10¹⁰⁰⁰.

Estu’s number = 9 × 10^1000003. Its exponent is much bigger.

So Estu’s number is greater.