Question 1(i)
Write 6×6×6×6 in exponential form.
Steps only
6×6×6×6 = 6⁴
Answer
6⁴
CLASS 8 · GANITA PRAKASH PART I · CHAPTER 2
Power Play
Original short answers and clear explanations for the chapter’s in-text activities and all numbered Figure it Out questions.
164 worked answersAll 14 end-of-chapter questions
164 worked answers · short steps first, explanations when you need them
Figure it Out — Exponential notation
Textbook pages 22–23
Question 1(ii)
Write y×y in exponential form.
Steps only
y×y = y²
Answer
y²
Question 1(iii)
Write b×b×b×b in exponential form.
Steps only
b×b×b×b = b⁴
Answer
b⁴
Question 1(iv)
Write 5×5×7×7×7 in exponential form.
Steps only
5×5×7×7×7 = 5²×7³
Answer
5²×7³
Question 1(v)
Write 2×2×a×a in exponential form.
Steps only
2×2×a×a = 2²a²
Answer
2²a²
Question 1(vi)
Write a×a×a×c×c×c×c×d in exponential form.
Steps only
a×a×a×c×c×c×c×d = a³c⁴d
Answer
a³c⁴d
Question 2(i)
Write the prime factorisation of 648 using powers.
Steps only
648 = 8×81
648 = 2³×3⁴
Answer
2³×3⁴
Question 2(ii)
Write the prime factorisation of 405 using powers.
Steps only
405 = 81×5
405 = 3⁴×5
Answer
3⁴×5
Question 2(iii)
Write the prime factorisation of 540 using powers.
Steps only
540 = 4×27×5
540 = 2²×3³×5
Answer
2²×3³×5
Question 2(iv)
Write the prime factorisation of 3600 using powers.
Steps only
3600 = 16×9×25
3600 = 2⁴×3²×5²
Answer
2⁴×3²×5²
Question 3(i)
Evaluate 2×10³.
Steps only
2×1000 = 2000
Answer
2000
Question 3(ii)
Evaluate 7²×2³.
Steps only
49×8 = 392
Answer
392
Question 3(iii)
Evaluate 3×4⁴.
Steps only
3×256 = 768
Answer
768
Question 3(iv)
Evaluate (−3)²×(−5)².
Steps only
9×25 = 225
Answer
225
Question 3(v)
Evaluate 3²×10⁴.
Steps only
9×10000 = 90000
Answer
90000
Question 3(vi)
Evaluate (−2)⁵×(−10)⁶.
Steps only
(−32)×1000000 = −32000000
Answer
−32000000
Figure it Out — Applying powers
Textbook pages 44–45
Question 1
Find the units digit of 2²²⁴ ÷ 4³².
Steps only
4³² = 2⁶⁴
2²²⁴÷2⁶⁴ = 2¹⁶⁰
units cycle: 2,4,8,6; 160 is a multiple of 4
Answer
6
Question 2
One new container of 5 bottles arrives each day. How many bottles arrive in 40 days?
Steps only
5×40 = 200 = 2×10²
Answer
200 bottles.
Question 3(i)
Write 64³ as a product of powers in three ways.
Steps only
64³ = 2¹⁸ = 2¹⁰×2⁸
= 4⁵×4⁴
= 8³×8³
Answer
Three possible forms are shown above.
Question 3(ii)
Write 192⁸ as a product of powers in three ways.
Steps only
192⁸ = (2⁶×3)⁸ = 2⁴⁸×3⁸
= 2⁴⁰×6⁸
= 2²⁴×2²⁴×3⁸
Answer
Three possible forms are shown above.
Question 3(iii)
Write 32⁻⁵ as a product of powers in three ways.
Steps only
32⁻⁵ = 2⁻²⁵ = 2⁻¹⁰×2⁻¹⁵
= 2⁻⁵×2⁻²⁰
= 4⁻¹²×2⁻¹
Answer
Three possible forms are shown above.
Question 4(i)
Classify as always, sometimes or never true: Cube numbers are also square numbers.
Steps only
64 = 4³ = 8²; 8 = 2³ is not a square.
Answer
Sometimes true.
Question 4(ii)
Classify as always, sometimes or never true: Fourth powers are square numbers.
Steps only
a⁴ = (a²)²
Answer
Always true.
Question 4(iii)
Classify as always, sometimes or never true: The fifth power of a number is divisible by its cube.
Steps only
a⁵/a³ = a², a ≠ 0
Answer
Always true for a nonzero integer.
Question 4(iv)
Classify as always, sometimes or never true: The product of two cubes is a cube.
Steps only
a³b³ = (ab)³
Answer
Always true.
Question 4(v)
Classify as always, sometimes or never true: q⁴⁶ is both a fourth power and a sixth power when q is prime.
Steps only
46 is not divisible by 4 or by 6.
Answer
Never true.
Question 5(i)
Simplify 10⁻²×10⁻⁵.
Steps only
10⁻²×10⁻⁵ = 10⁻⁷
Answer
10⁻⁷
Question 5(ii)
Simplify 5⁷÷5⁴.
Steps only
5⁷÷5⁴ = 5⁷⁻⁴ = 5³
Answer
5³
Question 5(iii)
Simplify 9⁻⁷÷9⁴.
Steps only
9⁻⁷÷9⁴ = 9⁻⁷⁻⁴ = 9⁻¹¹
Answer
9⁻¹¹
Question 5(iv)
Simplify (13⁻²)⁻³.
Steps only
(13⁻²)⁻³ = 13⁽⁻²⁾⁽⁻³⁾ = 13⁶
Answer
13⁶
Question 5(v)
Simplify m⁵n¹²(mn)⁹.
Steps only
m⁵n¹²m⁹n⁹ = m¹⁴n²¹
Answer
m¹⁴n²¹
Question 6(i)
If 12² = 144, find (1.2)².
Steps only
(1.2)² = 12²/10² = 144/100 = 1.44
Answer
1.44
Question 6(ii)
If 12² = 144, find (0.12)².
Steps only
(0.12)² = 12²/100² = 144/10000 = 0.0144
Answer
0.0144
Question 6(iii)
If 12² = 144, find (0.012)².
Steps only
(0.012)² = 12²/1000² = 144/1000000 = 0.000144
Answer
0.000144
Question 6(iv)
If 12² = 144, find (120)².
Steps only
120² = (12×10)² = 144×100 = 14400
Answer
14400
Question 7
Which of 2⁴×3⁶, 6⁴×3², 6¹⁰, 18²×6² and 6²⁴ are equal?
Steps only
2⁴×3⁶
6⁴×3² = 2⁴×3⁶
18²×6² = (2×3²)²(2×3)² = 2⁴×3⁶
6¹⁰ = 2¹⁰3¹⁰; 6²⁴ = 2²⁴3²⁴
Answer
2⁴×3⁶ = 6⁴×3² = 18²×6².
Question 8(i)
Which is greater: 4³ or 3⁴?
Steps only
4³ = 64; 3⁴ = 81
Answer
3⁴ is greater.
Question 8(ii)
Which is greater: 2⁸ or 8²?
Steps only
2⁸ = 256; 8² = 64
Answer
2⁸ is greater.
Question 8(iii)
Which is greater: 100² or 2¹⁰⁰?
Steps only
100² = 10000
2¹⁴ = 16384 > 10000; 2¹⁰⁰ > 2¹⁴
Answer
2¹⁰⁰ is greater.
Question 9
How many decimal digits are needed for unique IDs for 8.5 billion packets?
Steps only
10⁹ = 1 billion < 8.5 billion
10¹⁰ = 10 billion ≥ 8.5 billion
Answer
10 digits, allowing leading zeroes.
Question 10
Describe positive integers that are both squares and cubes.
Steps only
n⁶ = (n³)² = (n²)³
examples: 1,64,729,4096,…
Answer
Exactly the positive sixth powers.
Question 11
How many five-character codes use digits 0–9 and uppercase letters A–Z?
Steps only
choices per slot = 10+26 = 36
number of codes = 36⁵ = 60466176
Answer
60,466,176 codes.
Question 12
Using the textbook estimates of 10⁹ sheep and 10⁹ goats, find their combined population.
Steps only
10⁹+10⁹ = 2×10⁹
Answer
2×10⁹; select both 2×10⁹ and 10⁹+10⁹.
Question 13(i)
Estimate the number of clothing pieces if each of 8×10⁹ people has 30.
Steps only
(8×10⁹)×30 = 2.4×10¹¹
Answer
About 2.4×10¹¹ pieces.
Question 13(ii)
Estimate the bees in 100 million colonies with 50,000 bees each.
Steps only
(10⁸)×(5×10⁴) = 5×10¹²
Answer
About 5×10¹² bees.
Question 13(iii)
Estimate the bacterial population in all humans, using about 38 trillion bacteria per person and a world population of 8.2 billion.
Steps only
38×10¹²×8.2×10⁹ = 311.6×10²¹ = 3.116×10²³
Answer
About 3.116×10²³ bacterial cells.
Question 13(iv)
Estimate the time a person spends eating in a lifetime, in seconds.
Steps only
Example assumptions: 1.5 hours per day for 80 years.
1.5×3600×365×80 = 157680000 = 1.5768×10⁸
Answer
Example estimate: about 1.6×10⁸ seconds.
Question 14
How can you find the date one billion seconds before a chosen date?
Steps only
10⁹ seconds = 11574 days + 1 hour 46 minutes 40 seconds
Example: count backwards from 5 October 2026 at 00:00 to 26 January 1995, 22:13:20.
Answer
Example reference date: 5 October 2026. The result is 26 January 1995, 22:13:20.
Explore powers — In-text practice
Textbook pages 22–43
Question Thickness
A sheet starts at thickness v and is folded in half ten times. Find its new thickness.
Steps only
thickness = v×2¹⁰ = 1024v
Answer
2¹⁰v (option v).
Question Negative powers of one
Find (−1)⁵ and (−1)⁵⁶.
Steps only
(−1)⁵ = −1; (−1)⁵⁶ = 1
Answer
−1 and 1.
Question Even power
Check whether (−2)⁴ equals 16.
Steps only
(−2)(−2)(−2)(−2) = 4×4 = 16
Answer
Yes, (−2)⁴ = 16.
Question Zero base
What are 0², 0⁵ and 0ⁿ for positive integer n?
Steps only
0² = 0; 0⁵ = 0; 0ⁿ = 0 for n > 0
Answer
All are zero for positive n.
Question Rooms and diamonds
Count rooms and diamonds when there are 3 daughters, 3 baskets each, 3 keys per basket, 3 rooms per key, 3 tables per room, 3 necklaces per table and 3 diamonds per necklace.
Steps only
rooms = 3⁴ = 81
diamonds = 3⁷ = 2187
Answer
81 rooms and 2187 diamonds.
Question Pond transfer
Start with one lotus. Double daily for 4 days, then triple daily for 4 days. Does reversing the order change the final count?
Steps only
2⁴×3⁴ = (2×3)⁴ = 6⁴ = 1296
3⁴×2⁴ = 1296
Answer
1296 lotuses in either order.
Question Pond days
A pond starting with one lotus doubles daily and is full after 30 doublings. When is it half full?
Steps only
full: 2³⁰; half-full: 2²⁹
Answer
Day 29, one day before it is full.
Question Equal exponent product
Find 2⁵×5⁵.
Steps only
2⁵×5⁵ = (2×5)⁵ = 10⁵ = 100000
Answer
100000.
Question Equal exponent quotient
Simplify 10⁴/5⁴.
Steps only
(10/5)⁴ = 2⁴ = 16
Answer
2⁴ = 16.
Question Clothes
Choose from 7 dresses, 2 hats and 3 pairs of shoes. How many outfits are possible?
Steps only
7×2×3 = 42
Answer
42 outfits.
Question Letter lock
How many six-slot passwords use uppercase letters A–Z?
Steps only
26⁶ = 308915776
Answer
308,915,776 passwords.
Question Power quotient
Write 2¹⁰⁰÷2²⁵ as a power of 2.
Steps only
2¹⁰⁰⁻²⁵ = 2⁷⁵
Answer
2⁷⁵.
Question Nonzero base
Why must a base n be nonzero in nᵃ/nᵇ?
Steps only
If n = 0 and b > 0, the denominator is 0.
Answer
Division by zero is undefined.
Question Integer exponents
Do the exponent laws extend from positive exponents to all integer exponents?
Steps only
n⁰ = 1; n⁻ᵏ = 1/nᵏ, n ≠ 0
Answer
Yes, for nonzero bases.
Question Positive versus negative exponent
How many times as large is 4² as 4⁻²?
Steps only
4²/4⁻² = 4⁴ = 256
Answer
256 times.
Question A million seconds
Convert one million seconds to days.
Steps only
1000000/86400 = 11.574074… days
Answer
About 11.6 days, or roughly 12 days.
Question Grouping powers (1)
Compute 2⁹ by splitting it into a product of powers.
Steps only
2⁴×2⁵ = 16×32 = 512
Answer
512
Question Grouping powers (2)
Compute 5⁷ by splitting it into a product of powers.
Steps only
5³×5⁴ = 125×625 = 78125
Answer
78125
Question Grouping powers (3)
Compute 4⁶ by splitting it into a product of powers.
Steps only
4³×4³ = 64×64 = 4096
Answer
4096
Question Power of power (1)
Write 8⁶ as a power of a power in two ways.
Steps only
8⁶ = (8²)³ = (8³)²
Answer
(8²)³ = (8³)²
Question Power of power (2)
Write 7¹⁵ as a power of a power in two ways.
Steps only
7¹⁵ = (7³)⁵ = (7⁵)³
Answer
(7³)⁵ = (7⁵)³
Question Power of power (3)
Write 9¹⁴ as a power of a power in two ways.
Steps only
9¹⁴ = (9²)⁷ = (9⁷)²
Answer
(9²)⁷ = (9⁷)²
Question Power of power (4)
Write 5⁸ as a power of a power in two ways.
Steps only
5⁸ = (5²)⁴ = (5⁴)²
Answer
(5²)⁴ = (5⁴)²
Question Reciprocals (1)
Write 2⁻⁴ without a negative exponent.
Steps only
2⁻⁴ = 1/16
Answer
1/16
Question Reciprocals (2)
Write 10⁻⁵ without a negative exponent.
Steps only
10⁻⁵ = 1/100000
Answer
1/100000
Question Reciprocals (3)
Write (−7)⁻² without a negative exponent.
Steps only
(−7)⁻² = 1/49
Answer
1/49
Question Reciprocals (4)
Write (−5)⁻³ without a negative exponent.
Steps only
(−5)⁻³ = −1/125
Answer
−1/125
Question Reciprocals (5)
Write 10⁻¹⁰⁰ without a negative exponent.
Steps only
10⁻¹⁰⁰ = 1/10¹⁰⁰
Answer
1/10¹⁰⁰
Question Exponent practice (1)
Simplify 2⁻⁴×2⁷.
Steps only
2⁻⁴⁺⁷ = 2³
Answer
2³
Question Exponent practice (2)
Simplify 3²×3⁻⁵×3⁶.
Steps only
3²⁻⁵⁺⁶ = 3³
Answer
3³
Question Exponent practice (3)
Simplify p³×p⁻¹⁰.
Steps only
p³⁻¹⁰ = p⁻⁷, p ≠ 0
Answer
p⁻⁷ (p ≠ 0)
Question Exponent practice (4)
Simplify 2⁴×(−4)⁻².
Steps only
16×(1/16) = 1
Answer
1
Question Exponent practice (5)
Simplify 8ᵖ×8ᑫ.
Steps only
8ᵖ⁺ᑫ
Answer
8ᵖ⁺ᑫ
Question Place value (i)
Expand 172 using powers of 10.
Steps only
172 = 1×10²+7×10¹+2×10⁰
Answer
1×10²+7×10¹+2×10⁰
Question Place value (ii)
Expand 5642 using powers of 10.
Steps only
5642 = 5×10³+6×10²+4×10¹+2×10⁰
Answer
5×10³+6×10²+4×10¹+2×10⁰
Question Place value (iii)
Expand 6374 using powers of 10.
Steps only
6374 = 6×10³+3×10²+7×10¹+4×10⁰
Answer
6×10³+3×10²+7×10¹+4×10⁰
Question Standard form (i)
Write 59853 in scientific notation.
Steps only
59853 = 5.9853×10⁴
Answer
5.9853×10⁴
Question Standard form (ii)
Write 65950 in scientific notation.
Steps only
65950 = 6.595×10⁴
Answer
6.595×10⁴
Question Standard form (iii)
Write 3430000 in scientific notation.
Steps only
3430000 = 3.43×10⁶
Answer
3.43×10⁶
Question Standard form (iv)
Write 70040000000 in scientific notation.
Steps only
70040000000 = 7.004×10¹⁰
Answer
7.004×10¹⁰
Question Paper folding: thickness table
Complete the thickness pattern for folds 0 to 10 if the starting thickness is v.
Steps only
v, 2v, 4v, 8v, 16v, 32v, 64v, 128v, 256v, 512v, 1024v
Answer
After n folds the thickness is 2ⁿv.
Question Paper folding: large fold counts
How many folds are needed for the thickness to multiply by 8? By 1024?
Steps only
8 = 2³; 1024 = 2¹⁰
Answer
So 3 folds multiply the thickness by 8, and 10 folds multiply it by 1024.
Question Paper folding: thickness after 30 folds
If a paper is 0.001 cm thick, what thickness does the doubling model predict after 30 folds?
Steps only
0.001×2³⁰ = 0.001×1,073,741,824 = 1,073,741.824 cm
Answer
1,073,741.824
Question Paper folding: thickness after 45 folds
Using the same 0.001 cm starting thickness, what thickness does the doubling model predict after 45 folds?
Steps only
0.001×2⁴⁵ = 35,184,372,088.832 cm
Answer
35,184,372,088.832
Question Paper folding: compare growth
How many times thicker is a sheet after 10 folds than after 3 folds?
Steps only
2¹⁰v÷2³v = 2⁷ = 128
Answer
It is 128 times as thick.
Question The Stones that Shine
In a story, each stage triples the number of stones. How many stones are there after 7 stages if there is 1 at the start?
Steps only
1×3⁷ = 2187
Answer
There are 2,187 stones after seven tripling stages.
Question The Stones that Shine: exponent split
Show that 3⁷ can be written as 3⁴×3³ in two ways.
Steps only
3⁴×3³ = 3⁴⁺³ = 3⁷
Answer
The factors can be grouped into four and three of the seven equal factors.
Question Magical pond: full and half covered
A pond’s lotus count doubles each day and is full on day 30. Write the count as a power of 2 when full and when half full.
Steps only
Full: 2³⁰; half full: 2²⁹
Answer
The pond is half full on day 29.
Question Combination lock: five digits
How many different five-digit codes can be made using digits 0–9 if repetition is allowed?
Steps only
10×10×10×10×10 = 10⁵ = 100,000
Answer
There are 100,000 codes.
Question Combination lock: six letters
A lock has six places and each place can be any of 26 letters. How many passwords are possible?
Steps only
26⁶ = 308,915,776
Answer
There are 308,915,776 possible passwords.
Question Linear growth and exponential growth
Explain the difference between adding 20 cm at every step and doubling a quantity at every stage.
Steps only
Linear: add the same amount; exponential: multiply by the same factor.
Answer
Adding 20 cm gives equal increases. Doubling makes each increase depend on the current amount, so the growth quickly becomes much faster.
Question Ladder to the Moon
A ladder has steps 20 cm apart. About how many steps would span 384,400 km to the Moon?
Steps only
20 cm = 0.0002 km; 384,400÷0.0002 = 1,922,000,000
Answer
About 1.922 billion steps are needed.
Question Powers of ten: large quantities
Write 30,000,000,000,000,000,000 m in scientific notation.
Steps only
30,000,000,000,000,000,000 = 3×10¹⁹
Answer
3×10¹⁹ m.
Question Powers of ten: stars
Write 100,000,000,000 stars in scientific notation.
Steps only
100,000,000,000 = 1×10¹¹
Answer
1×10¹¹
Question Powers of ten: Earth mass
Write 5,976,000,000,000,000,000,000,000 kg in scientific notation.
Steps only
5,976,000,000,000,000,000,000,000 = 5.976×10²⁴
Answer
The mass is 5.976×10²⁴ kg.
Question Earth on the Sun–Saturn number line
The Earth–Sun distance is 1.496×10¹¹ m and the Saturn–Sun distance is 1.4335×10¹² m. Where does Earth lie relative to Saturn?
Steps only
(1.496×10¹¹)÷(1.4335×10¹²) ≈ 0.104
Answer
Earth is about 0.104 of the way from the Sun to Saturn, or roughly one-tenth of that distance.
Question Estimate a donation: jaggery
Estimate the value of jaggery equal to Roxie’s weight. Use a reasonable example assumption.
Steps only
45×70 = ₹3,150
Answer
3,150
Question Estimate a donation: wheat
Estimate the value of wheat equal to Estu’s weight. Use a reasonable example assumption.
Steps only
50×50 = ₹2,500
Answer
2,500
Question Estimate coins equal to body weight
About how many ₹1 coins would equal a person’s body weight? State an assumption.
Steps only
45,000÷5 = 9,000
Answer
9,000
Question Estimate five-rupee coins
Using the same assumptions, about how much money would ₹5 coins weighing as much as a 45 kg person represent? Assume each coin weighs 6 g.
Steps only
45,000÷6 = 7,500 coins; 7,500×₹5 = ₹37,500
Answer
₹37,500
Question Estimate notebooks and meals
Estimate how many people could benefit if a person donates notebooks or meals worth their body weight. State assumptions.
Steps only
50×₹50 = ₹2,500; ₹2,500÷₹25 = 100 meals
Answer
About 100 meals.
Question Estimate a 400 km walking journey
A 400 km journey ends early this morning. If someone walks 5 km per hour for 8 hours each day, about how many days would the journey take?
Steps only
5×8 = 40 km/day; 400÷40 = 10 days
Answer
It would take about 10 walking days, so the journey began roughly 10 days ago.
Question Estimate one billion seconds
About how many years are in one billion seconds?
Steps only
1,000,000,000÷(365×24×60×60) ≈ 31.7
Answer
31.7
Question Estimate lifetime walking distance
If someone walks 5 km per hour for 2 hours each day over 70 years, about how many times could they walk around Earth? Use 40,000 km for Earth’s circumference.
Steps only
5×2×365×70 = 255,500 km; 255,500÷40,000 ≈ 6.4
Answer
About 6 times around Earth, using these assumptions.
Question Date one billion seconds earlier
How can you find the date one billion seconds before a chosen date?
Steps only
1,000,000,000 seconds = 11,574 days, 1 hour, 46 minutes, 40 seconds
Answer
The exact calendar date depends on the chosen starting date and leap years.
Question Sun–Saturn distances
Which is smaller: the Sun–Saturn distance 1.4335×10¹² m, the Saturn–Uranus distance 1.439×10¹² m, or the Sun–Earth distance 1.496×10¹¹ m?
Steps only
1.496×10¹¹ < 1.4335×10¹² < 1.439×10¹²
Answer
The Sun–Earth distance is smallest.
Question Age in hours
Roxie is 4,840 days old. About how many hours old is she?
Steps only
4,840×24 = 116,160
Answer
Roxie is 116,160 hours old.
Question Estimate journey to the Moon with paper folds
A paper starts 0.001 cm thick. How many times must its thickness double to reach approximately 384,400 km?
Steps only
384,400 km = 38,440,000,000 cm; 0.001×2⁴⁶ ≈ 70,369,744,177 cm
Answer
46
Question Examples of growth
Give one example of linear growth and one example of exponential growth.
Steps only
Linear: save ₹10 each week. Exponential: a quantity doubles each day.
Answer
For example, saving ₹10 each week is linear; a quantity that doubles each day grows exponentially.
Question Large numbers: starlings
Write an estimated population of 1.3 billion starlings in scientific notation.
Steps only
1.3 billion = 1.3×10⁹
Answer
About 1.3×10⁹ starlings.
Question Large numbers: mosquitoes
Write an estimated population of 110 trillion mosquitoes in scientific notation.
Steps only
110 trillion = 110×10¹² = 1.1×10¹⁴
Answer
About 1.1×10¹⁴ mosquitoes.
Question Large numbers: people per elephant
Using 8×10⁹ people and 4×10⁵ African elephants, about how many people are there per elephant?
Steps only
(8×10⁹)÷(4×10⁵) = 2×10⁴ = 20,000
Answer
About 20,000 people per elephant.
Question Dinosaur fossil age
A fossil is about 15 million years old. Write this age in seconds using scientific notation, using 1 year ≈ 3.15×10⁷ seconds.
Steps only
1.5×10⁷×3.15×10⁷ = 4.725×10¹⁴ seconds
Answer
About 4.725×10¹⁴ seconds.
Question Plants on land age
Land plants appeared about 470 million years ago. Convert this to seconds using 1 year ≈ 3.15×10⁷ seconds.
Steps only
4.7×10⁸×3.15×10⁷ = 1.4805×10¹⁶ seconds
Answer
About 1.48×10¹⁶ seconds.
Question Counting stars
If one star is counted each second, how long would it take to count 2×10²³ stars?
Steps only
2×10²³ stars × 1 second/star = 2×10²³ seconds
Answer
2×10²³ seconds.
Question Drinking Earth’s water
At one 200 ml glass every 10 seconds, how long would it take to drink 1.25×10²⁴ ml of water?
Steps only
(1.25×10²⁴÷200)×10 = 6.25×10²² seconds
Answer
6.25×10²² seconds, under this thought-experiment rate.
Question Large powers naming pattern
What does the prefix in names such as million, billion and trillion show in the international system?
Steps only
million = 10⁶; billion = 10⁹; trillion = 10¹²
Answer
The prefix marks how many groups of three zeros, or factors of 1,000, are involved.
Question Historical number names
If each next historical number name is one hundred times the previous one, by what factor does the value increase at each step?
Steps only
×100 = ×10²
Answer
Each step multiplies the value by 100, or 10².
Question Power contest: compare expressions
Which is greater: 4×10¹⁰⁰ or 9,000×10⁹⁹?
Steps only
9,000×10⁹⁹ = 0.9×10¹⁰⁰; 4×10¹⁰⁰ > 0.9×10¹⁰⁰
Answer
4×10¹⁰⁰
Question Roxie’s age in days
Roxie is 4,840 days old. About how many years is this?
Steps only
4,840÷365 ≈ 13.26 years
Answer
13.26
Question Estu date of birth from days
How can you find Estu’s date of birth if Estu is 4,070 days old today?
Steps only
4,070 days is about 11 years and 2 months.
Answer
The exact birth date depends on the stated “today”; subtract 4,070 calendar days from that date.
Question What could 6,970,710 mean?
Roxie says “I am 6,970,710 … old.” Give a sensible unit this number could represent.
Steps only
6,970,710 seconds÷86,400 ≈ 80.7 days
Answer
80.7 days
Question Billions of time comparison
Write 15 million years in seconds in scientific notation.
Steps only
15×10⁶×3.15×10⁷ = 4.725×10¹⁴ seconds
Answer
About 4.725×10¹⁴ seconds.
Question In-text: Thickness after two folds
After two folds, by how much has the paper’s thickness increased compared with its starting thickness?
Steps only
2² = 4 times as thick; increase = 4v−v = 3v
Answer
It is 4 times as thick, which is an increase of 3v.
Question In-text: Thickness table: 10 folds
A paper begins at 0.001 cm. Find the modelled thickness after 10 folds.
Steps only
0.001×2¹⁰ = 0.001×1024 = 1.024 cm
Answer
1.024 cm.
Question In-text: Thickness table: 20 folds
A paper begins at 0.001 cm. Find the modelled thickness after 20 folds.
Steps only
0.001×1,048,576 = 1,048.576 cm = 10.48576 m
Answer
About 10.486 m.
Question In-text: Thickness table: 30 folds exact
A paper begins at 0.001 cm. Find the modelled thickness after 30 folds.
Steps only
0.001×1,073,741,824 cm = 1,073,741.824 cm = 10.7374 km
Answer
About 10.737 km.
Question In-text: Thickness table: 40 folds
A paper begins at 0.001 cm. Find the modelled thickness after 40 folds.
Steps only
0.001×1,099,511,627,776 cm ≈ 10,995.1 km
Answer
About 10,995 km.
Question In-text: Fold interval multiplier
By what factor does paper thickness increase between fold 10 and fold 20?
Steps only
2¹⁰ = 1024
Answer
It increases by a factor of 1,024.
Question In-text: Fold interval multiplier 20–30
By what factor does paper thickness increase between fold 20 and fold 30?
Steps only
2¹⁰ = 1024
Answer
It increases by a factor of 1,024.
Question In-text: Fold interval multiplier 30–40
By what factor does paper thickness increase between fold 30 and fold 40?
Steps only
2¹⁰ = 1024
Answer
It increases by a factor of 1,024.
Question In-text: Prime factorisation of 32400
Write 32,400 as a product of prime powers.
Steps only
32,400 = (2²×3⁴)×(2²×5²) = 2⁴×3⁴×5²
Answer
2⁴×3⁴×5².
Question In-text: Power multiplication: p⁴×p⁶
Write p⁴×p⁶ as one power.
Steps only
p⁴×p⁶ = p⁴⁺⁶ = p¹⁰
Answer
p¹⁰.
Question In-text: Compute 2⁹
Compute 2⁹ by splitting it into two powers.
Steps only
2⁹ = 2⁴×2⁵ = 16×32 = 512
Answer
512.
Question In-text: Compute 5⁷
Compute 5⁷ by splitting it into two powers.
Steps only
5⁷ = 5³×5⁴ = 125×625 = 78,125
Answer
78,125.
Question In-text: Compare 4² and 4⁻²
How many times as large is 4² as 4⁻²?
Steps only
4²÷4⁻² = 4⁴ = 256
Answer
4² is 256 times as large.
Question In-text: Power line: 4⁸÷4³
Use exponent rules to find 4⁸÷4³.
Steps only
4⁸÷4³ = 4⁵ = 1024
Answer
1,024.
Question In-text: Power line: 4³×4²
Use exponent rules to find 4³×4².
Steps only
4³×4² = 4⁵ = 1024
Answer
1,024.
Question In-text: Power line: 4⁷×4⁻²
Simplify 4⁷×4⁻².
Steps only
4⁷×4⁻² = 4⁵ = 1024
Answer
1,024.
Question In-text: Power line: 7²⁰⁴⁰?
Find 2401×49 using powers of 7.
Steps only
7⁴×7² = 7⁶ = 117,649
Answer
117,649.
Question In-text: Power line: 49³
Evaluate 49³ using powers of 7.
Steps only
49³ = (7²)³ = 7⁶ = 117,649
Answer
117,649.
Question In-text: Power line: 343×2401
Evaluate 343×2401 using powers of 7.
Steps only
7³×7⁴ = 7⁷ = 823,543
Answer
823,543.
Question In-text: Power line: 16807÷49
Evaluate 16807÷49 using powers of 7.
Steps only
7⁵÷7² = 7³ = 343
Answer
343.
Question In-text: Power line: reciprocal multiplication
Evaluate (1/343)×(1/343) using powers of 7.
Steps only
7⁻³×7⁻³ = 7⁻⁶ = 1/117649
Answer
1/117,649.
Question In-text: Decimal place-value powers
Write 561.903 using powers of 10.
Steps only
5×10² + 6×10¹ + 1×10⁰ + 9×10⁻¹ + 0×10⁻² + 3×10⁻³
Answer
5×10²+6×10+1+9×10⁻¹+3×10⁻³.
Question In-text: Human population in billions
Write a world population of about 8.2 billion using powers of 10.
Steps only
8.2 billion = 8.2×10⁹
Answer
8.2×10⁹ people.
Question In-text: Compare distance to Uranus
Which is smaller: Saturn–Uranus 1.439×10¹² m or Sun–Earth 1.496×10¹¹ m?
Steps only
1.496×10¹¹ < 1.439×10¹²
Answer
Sun–Earth is smaller.
Question In-text: Moon distance in paper sheets
If one sheet is 0.001 cm thick, how many stacked sheets make 384,400 km?
Steps only
384,400 km = 38,440,000,000,000 cm; ÷0.001 cm = 3.844×10¹⁶ sheets
Answer
3.844×10¹⁶ sheets.
Question In-text: Roxie age in hours
Roxie is 4,840 days old. How many hours old is she?
Steps only
4,840×24 = 116,160 hours
Answer
116,160 hours.
Question In-text: Estu date of birth
Estu is 4,070 days old today. How do you find the date of birth?
Steps only
4,070 days is about 11 years and 2 months; exact date requires the reference date.
Answer
Subtract 4,070 calendar days from the stated current date.
Question In-text: Lifetime eating estimate
Estimate total time spent eating in a lifetime. Use 1.5 hours daily for 80 years.
Steps only
1.5×365×80×3600 = 157,680,000 seconds ≈ 1.58×10⁸
Answer
About 1.58×10⁸ seconds.
Question In-text: Count all ants versus people
Using 2×10¹⁶ ants and 8×10⁹ people, how many ants are there per person?
Steps only
(2×10¹⁶)÷(8×10⁹) = 0.25×10⁷ = 2.5×10⁶
Answer
About 2.5 million ants per person.
Question In-text: Starlings flocks
If there are 1.3×10⁹ starlings and each flock has 10,000 birds, how many flocks is that?
Steps only
1.3×10⁹÷10⁴ = 1.3×10⁵
Answer
About 130,000 flocks.
Question In-text: Leaves on trees
There are about 3×10¹² trees. If each has 10⁴ leaves, how many leaves are there?
Steps only
3×10¹²×10⁴ = 3×10¹⁶
Answer
About 3×10¹⁶ leaves.
Question In-text: Compare Earth and Saturn distance
How far along the Sun–Saturn line is Earth if Earth is 1.496×10¹¹ m away and Saturn is 1.4335×10¹² m away?
Steps only
1.496×10¹¹÷1.4335×10¹² ≈ 0.104
Answer
About 10.4% of the way from Sun to Saturn.
Question In-text: Earth water volume thought experiment
At one 200 ml glass every 10 seconds, how long to drink 1.25×10²² litres of Earth’s water?
Steps only
1.25×10²² L = 1.25×10²⁵ ml; ÷200×10 = 6.25×10²³ seconds
Answer
6.25×10²³ seconds.