Power Play

Each fold DOUBLES the thickness, so after 10 folds we multiply the initial thickness v by 2, ten times over — that's 2¹⁰ multiplied by v.

(v) 2¹⁰v is correct. (iv) is close but forgets to multiply by v; the others (10v, 10+v, 2×10×v, 10²v) all describe ADDING or a fixed multiple, not repeated doubling.

Split each exponent into two smaller pieces that add up to it, then multiply the two smaller powers:

There's more than one valid split for each one — e.g. 2⁹ = 2³×2⁶ works too. As long as the two exponents add up to the original, the answer comes out the same!

Keep dividing by the smallest prime that fits, until you reach 1:

32400 = 2⁴ × 5² × 3⁴

A negative number multiplied an ODD number of times stays negative; an EVEN number of times, the minus signs cancel in pairs and it turns positive.

(−1)⁵ = −1 (negative, since 5 is odd). (−1)⁵⁶ = 1 (positive, since 56 is even).

Yes, (−2)⁴ = 16. Four negatives multiplied together (an even count) give a positive answer.

0ⁿ = 0, for any n > 0 — multiplying zero by itself any number of times still gives zero! (The one exception is 0⁰, which is left undefined — that's actually WHY we said n can't be 0 back when we defined n⁰ = 1.)

Watch the signs! An odd power of a negative number stays negative.

Just subtract the exponents (100 − 25 = 75) — no need to actually multiply out either giant number!

Using n⁻ᵃ = 1/nᵃ — just "flip" the base to the bottom of a fraction and make the exponent positive:

Same base multiplying? ADD the exponents — even when some of them are negative or letters!

Yes! Since 4⁷ ÷ 4⁵ = 4², and 16384 ÷ 1024 = 16 = 4². ✓

4² is 4⁴ times larger than 4⁻² (because 4² ÷ 4⁻² = 4⁽²⁻⁽⁻²⁾⁾ = 4⁴ = 256).

561.903 = (5×100) + (6×10) + 1 + (9×1/10) + (0×1/100) + (3×1/1000). Writing every part as a power of 10:

Notice how the negative exponents count DOWN past zero, exactly like the Power Lines you saw earlier!

This is exactly WHY scientific notation exists — counting all those zeros by eye is a recipe for mistakes!

Compare that to just 46 FOLDS of a single sheet reaching the Moon — that's the incredible power of doubling (exponential growth) versus simply stacking sheets one at a time!

Every day has 24 hours, so multiply:

Count 4070 days backward from today. 4070 days is 4070 ÷ 365.25 ≈ 11.1 years, so go back about 11 years, then fine-tune day by day:

So Estu's date of birth works out to roughly 9 July 2015 (using today, 30 Aug 2026, as the reference date — your exact answer will shift depending on what "today" is when you solve it!).

A ₹1 coin weighs about 4–5 grams (say 4.5 g = 0.0045 kg). Number of coins needed:

That's TEN THOUSAND coins just to match one 13-year-old's weight! (Your answer may differ slightly depending on the coin weight you assume — that's fine.)

Assume an adult weighs ≈60 kg. If each notebook weighs ≈100 g (0.1 kg) and goes to one student:

For annadāna, if one meal weighs ≈0.5 kg: 60 ÷ 0.5 = 120 meals — enough to feed roughly 120 people that year. (Again — reasonable assumptions, your numbers may vary!)

Assume a comfortable walking pace of about 20 km/day (walking several hours, with rest):

So they'd have set out roughly 20 days earlier. (If they walked faster — say 25–30 km/day — it could be closer to 13–16 days; estimates like this depend on your assumed pace.)

1. Guess first — would it be less than once around? A few times? Dozens of times?

2. Model: Total distance walked = walking speed × hours walked per day × days walked per year × number of years. Then divide by 40,000 km to get the number of times around the Earth.

3. Compute — assuming a steady walking pace of ≈5 km/h, for about 8 hours a day, on roughly 300 days a year (allowing for rest, bad weather etc.), over a 60-year walking "career":

So a person could walk around the Earth roughly 18 times in a lifetime! (Your own assumptions about pace, hours, and years may reasonably give something a bit different — that's fine, as long as your reasoning is sound.)

Each prefix is a Latin/Greek counting word: quad=4, quint=5, sext=6, sept=7, oct=8, non=9, dec=10.

It tells you how many groups of "thousand" (10³) are multiplied on top of a million (10⁶). E.g. quadrillion = 10⁶ × (10³)⁴⁻¹ = 10¹⁵ — the prefix number is basically the count of thousands beyond the first one.

India's highest-value note today is ₹2000. But history has seen far wilder numbers:

• Hungary, 1946: a note valued 1 sextillion pengő (10²¹, called "1 milliard b.-pengő") was printed but never actually issued.
• Zimbabwe, 2009: a real, issued 100 trillion (10¹⁴) Zimbabwean dollar note — worth only about $30 at the time it was printed, due to hyperinflation!

Both are dramatic real-world reminders of why scientific notation matters — these are numbers no one could safely write out digit by digit!

One new container of 5 bottles arrives EACH day — so after 40 days there are 40 containers, each holding 5 bottles. This is ADDING containers (not multiplying), so:

64 = 2⁶, so 64³ = 2¹⁸. Splitting the exponent 18 different ways (or splitting the base):

There are lots of correct ways to split these — as long as multiplying the pieces back together gives the original number!

Each one has more than one right answer — just make sure the two exponents in your split MULTIPLY to give the original power (e.g. for 7¹⁵: 3×5=15 either way)!

10⁹ = 1,000,000,000 — that's LESS than 8.5 billion, so 9 digits isn't enough.

At least 10 digits are needed.

Yes, infinitely many! They're exactly the 6th powers (n⁶), since n⁶ = (n³)² = (n²)³ — always both a square AND a cube at once. Examples: 1, 64, 729, 4096…

(i) Only Sometimes True. It's true only for 6th powers (like 64=4³=8²), but 27=3³ is a cube that is NOT a square.

(ii) Always True. A fourth power n⁴ = (n²)² — which is always a square number.

(iii) Always True. Since n⁵ ÷ n³ = n⁵⁻³ = n², a whole number — so n⁵ always divides evenly by n³.

(iv) Always True. (n₁)³ × (n₂)³ = (n₁×n₂)³ — still a perfect cube.

(v) Never True. q⁴⁶ is a 4th power only if 46 is divisible by 4, and a 6th power only if 46 is divisible by 6 — but 46÷4=11.5 and 46÷6≈7.67, neither divides evenly, so q⁴⁶ (q prime) is NEVER both.

Rewrite 4³² as 2⁶⁴, so:

Units digits of powers of 2 repeat in a cycle of 4: 2,4,8,6, 2,4,8,6… Since 160 ÷ 4 = 40 exactly, 2¹⁶⁰ has the same units digit as 2⁴ (=16).

Units digit = 6

2⁴×3⁶, 6⁴×3², and 18²×6² are all equal (all = 11,664) — because rewriting each in terms of the SAME prime factors (2 and 3) shows they're really the same expression in disguise!

Don't assume the bigger base always wins — the size of the EXPONENT usually matters much more, especially as numbers grow!

10⁹ + 10⁹ = 2×10⁹ — you ADD separate populations together (multiplying would give the wrong, absurdly large 10¹⁸, which makes no real-world sense here)!

A million seconds is barely more than a week and a half! But a BILLION seconds is about 31.7 years — that's the wild jump from million to billion.

Roxie's sum is only about 4 × 10¹⁰⁰⁰ — still "sized" like 10¹⁰⁰⁰ (adding four copies barely changes the size of such a huge number). Estu's number has exponent 1,000,000 — a MUCH bigger power of 10, multiplied by a modest 9000.

Estu wins by an enormous margin — 10¹⁰⁰⁰⁰⁰⁰ dwarfs 10¹⁰⁰⁰ (a million-digit exponent versus a thousand-digit one)!

(b) 1

(b) 1 ≤ x < 10

Every time you move the decimal point in the base by one place, the SQUARE's decimal point moves by TWO places (since it's squared):

(i) Clothing items:

(ii) Honeybees (100 million colonies = 10⁸, 50,000 bees each):

(iii) Bacterial cells worldwide (38 trillion per person, ≈8.2×10⁹ people):

(iv) Lifetime eating time (assume ~70-year lifespan, ~1 hour of eating per day):

1 arab / 1 billion seconds = 1×10⁹ ÷ 86,400 seconds-per-day ≈ 11,574 days ≈ 31.7 years.

So exactly 1 billion seconds before today (30 August 2026) lands on approximately 21 December 1994 — the book's official answer just says "≈31.7 years ago," so this is our best-effort exact date working from today.

(a) With repeats allowed: 10⁵ = 1,00,000 codes.

(b) All-different digits: 10×9×8×7×6 = 30,240 codes.

(c) Repeats-allowed (10⁵) is MORE secure — it gives far more possible codes (1,00,000 vs 30,240), making it harder to guess.