2 POWER PLAY
2.1 Experiencing the Power Play ...
An Impossible Venture!
Take a sheet of paper, as large a sheet as you can find. Fold it once. Fold
it again, and again.
How many times can you fold it over and over?
Estu says “I heard that a sheet of paper can’t be folded more than
7 times”.
Roxie replies “What if we use a thinner paper, like a newspaper or a
tissue paper?”
Try it with different types of paper and see what happens.
What! That’s crazy!
If you can fold a paper Just 46 times!? You
46 times, it will be so must have ignored
thick that it can reach Well, why
several zeros
the Moon! don’t you find
after 46.
out yourselves.
Say you can fold a sheet of paper as many times as you wish. What would
its thickness be after 30 folds? Make a guess.
Let us find out how thick a sheet of paper will be after 46 folds. Assume
that the thickness of the sheet is 0.001 cm.
Reprint 2026-27
, Ganita Prakash | Grade 8
The following table lists the thickness after each fold. Observe that the
thickness doubles after each fold.
Fold Thickness Fold Thickness Fold Thickness
1 0.002 cm 7 0.128 cm 13 8.192 cm
2 0.004 cm 8 0.256 cm 14 16.384 cm
3 0.008 cm 9 0.512 cm 15 32.768 cm
4 0.016 cm 10 1.024 cm 16 65.536 cm
5 0.032 cm 11 2.048 cm 17 ≈ 131 cm
6 0.064 cm 12 4.096 cm
(We use the sign ‘≈’ to indicate ‘approximately equal to’.)
After 10 folds, the thickness is just above 1 cm (1.024 cm).
After 17 folds, the thickness is about 131 cm (a little more than 4 feet).
Now, what do you think the thickness would be after 30 folds? Math
45 folds? Make a guess. Talk
Fill the table below.
Fold Thickness Fold Thickness Fold Thickness
18 ≈ 262 cm 21 24
19 ≈ 524 cm 22 25
20 ≈ 10.4 m 23 26
After 26 folds, the thickness is approximately 670 m. Burj Khalifa
in Dubai, the tallest building in the world, is 830 m tall.
Fold Thickness Fold Thickness
27 ≈ 1.3 km 29
28 30
After 30 folds, the thickness of the paper is about 10.7 km, the
typical height at which planes fly. The deepest point discovered in
the oceans is the Mariana Trench, with a depth of 11 km.
Fold Thickness Fold Thickness Fold Thickness
31 36 41
32 37 42
33 38 43
34 39 44
35 40 45
20
Reprint 2026-27
, Power Play
It might be hard to digest the fact that after just 46 folds, the thickness
is more than 7,00,000 km. This is the power of multiplicative growth,
also called exponential growth. Let us analyse the growth.
We have seen that the thickness doubles after every fold.
Fold 4 0.016 cm Fold 9 0.512 cm
Fold 5 0.032 cm Fold 10 1.024 cm
Notice the change in thickness after two
Fold 4 0.016 cm
folds. By how much does it increase?
After any 3 folds, the thickness increases 8 Fold 6 0.064 cm
times (= 2 × 2 × 2). Check if that is true. Similarly,
from any point, the thickness after 10 folds increases by 1024 times
(= 2 multiplied by itself 10 times), as shown in the table below.
Fold Thickness Times increased by
0 to 10 1.024 cm – 0.001 cm 1.024 ÷ 0.001
= 1.023 cm = 1024
10 to 20 10.485 m – 1.024 cm 10.485 m ÷ 1.024 cm
≈ 10.474 m = 1024
20 to 30 10.737 km – 10.485 m 10.737 km ÷ 10.485 m
≈ 10.726 km = 1024
30 to 40 10995 km – 10.737 km 10995 km ÷
≈ 10984.2 km 10.737 km = 1024
2.2 Exponential Notation and Operations
The initial thickness of the paper was 0.001 cm.
Upon folding once, its thickness became 0.001 cm × 2 = 0.002 cm.
Folding it twice, its thickness became —
0.001 cm × 2 × 2 = 0.004 cm, or 0.001 cm × 22 = 0.004 cm (in shorthand).
Upon folding it thrice, its thickness became —
0.001 cm × 2 × 2 × 2, or 0.001 cm × 23 = 0.008 cm.
When folded four times, its thickness became —
0.001 cm × 2 × 2 × 2 × 2, or 0.001 cm × 24 = 0.016 cm.
Similarly, the expression for the thickness of the paper when folded 7
times will be 0.001 cm × 2 × 2 × 2 × 2 × 2 × 2 × 2, or 0.001 cm × 27 = 0.128 cm.
We have seen that square numbers can be expressed as n2 and cube
numbers as n3.
n × n = n2 (read as ‘n squared’ or ‘n raised to the power 2’)
21
Reprint 2026-27
, Ganita Prakash | Grade 8
n × n × n = n3 (read as ‘n cubed’ or ‘n raised to the power 3’)
n × n × n × n = n4 (read as ‘n raised to the power 4’ or ‘the 4th power of n’)
n × n × n × n × n × n × n = n7 (read as ‘n raised to the power 7’ or ‘the 7th
power of n’) and so on.
In general, we write na to denote n multiplied by itself a times.
54 = 5 × 5 × 5 × 5 = 625.
54 is the exponential form of 625. Here, 4 is the
exponent/power, and 5 is the base. Exponents of 5 is read as
4
the form 5n are called powers of 5: 51, 52, 53, 54, etc. ‘5 raised to the power 4’ or
‘5 to the power 4’ or
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 210 = 1024.
‘5 power 4’ or
Remember the 1024 from earlier? There, it meant ‘4th power of 5’
that after every 10 folds, the thickness increased
1024 times.
Which expression describes the thickness of a sheet of paper after
it is folded 10 times? The initial thickness is represented by the
letter-number v.
(i) 10v (ii) 10 + v (iii) 2 × 10 × v
(iv) 210 (v) 210v (vi) 102v
Some more examples of exponential notation:
4 × 4 × 4 = 43 = 64.
(– 4) × (– 4) × (– 4) = (– 4)3= – 64.
Similarly,
a × a × a × b × b can be expressed as a3b2 (read as a cubed b squared).
a × a × b × b × b × b can be expressed as a2b4 (read as a squared b raised
to the power 4).
Remember that 4 + 4 + 4 = 3 × 4 = 12, whereas 4 × 4 × 4 = 43 = 64.
Express the number 32400 as a product of its prime factors
2 32400
and represent the prime factors in their exponential form.
2 16200
32400 = 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 × 3 × 3. 2 8100
In exponential form, this would be 2 4050
32400 = 24 × 52 × 34. 5 2025
5 405
What is (– 1)5 ? Is it positive or negative? What about (– 1)56 ?
3 81
Is (– 2)4 = 16? Verify. What is 02, 05 ? 3 27
What is 0n ? 3 9
Figure it Out
3 3
1. Express the following in exponential form:
1
(i) 6×6×6×6 (ii) y×y
(iii) b×b×b×b (iv) 5×5×7×7×7
(v) 2×2×a×a (vi) a×a×a×c×c×c×c×d
22
Reprint 2026-27
2.1 Experiencing the Power Play ...
An Impossible Venture!
Take a sheet of paper, as large a sheet as you can find. Fold it once. Fold
it again, and again.
How many times can you fold it over and over?
Estu says “I heard that a sheet of paper can’t be folded more than
7 times”.
Roxie replies “What if we use a thinner paper, like a newspaper or a
tissue paper?”
Try it with different types of paper and see what happens.
What! That’s crazy!
If you can fold a paper Just 46 times!? You
46 times, it will be so must have ignored
thick that it can reach Well, why
several zeros
the Moon! don’t you find
after 46.
out yourselves.
Say you can fold a sheet of paper as many times as you wish. What would
its thickness be after 30 folds? Make a guess.
Let us find out how thick a sheet of paper will be after 46 folds. Assume
that the thickness of the sheet is 0.001 cm.
Reprint 2026-27
, Ganita Prakash | Grade 8
The following table lists the thickness after each fold. Observe that the
thickness doubles after each fold.
Fold Thickness Fold Thickness Fold Thickness
1 0.002 cm 7 0.128 cm 13 8.192 cm
2 0.004 cm 8 0.256 cm 14 16.384 cm
3 0.008 cm 9 0.512 cm 15 32.768 cm
4 0.016 cm 10 1.024 cm 16 65.536 cm
5 0.032 cm 11 2.048 cm 17 ≈ 131 cm
6 0.064 cm 12 4.096 cm
(We use the sign ‘≈’ to indicate ‘approximately equal to’.)
After 10 folds, the thickness is just above 1 cm (1.024 cm).
After 17 folds, the thickness is about 131 cm (a little more than 4 feet).
Now, what do you think the thickness would be after 30 folds? Math
45 folds? Make a guess. Talk
Fill the table below.
Fold Thickness Fold Thickness Fold Thickness
18 ≈ 262 cm 21 24
19 ≈ 524 cm 22 25
20 ≈ 10.4 m 23 26
After 26 folds, the thickness is approximately 670 m. Burj Khalifa
in Dubai, the tallest building in the world, is 830 m tall.
Fold Thickness Fold Thickness
27 ≈ 1.3 km 29
28 30
After 30 folds, the thickness of the paper is about 10.7 km, the
typical height at which planes fly. The deepest point discovered in
the oceans is the Mariana Trench, with a depth of 11 km.
Fold Thickness Fold Thickness Fold Thickness
31 36 41
32 37 42
33 38 43
34 39 44
35 40 45
20
Reprint 2026-27
, Power Play
It might be hard to digest the fact that after just 46 folds, the thickness
is more than 7,00,000 km. This is the power of multiplicative growth,
also called exponential growth. Let us analyse the growth.
We have seen that the thickness doubles after every fold.
Fold 4 0.016 cm Fold 9 0.512 cm
Fold 5 0.032 cm Fold 10 1.024 cm
Notice the change in thickness after two
Fold 4 0.016 cm
folds. By how much does it increase?
After any 3 folds, the thickness increases 8 Fold 6 0.064 cm
times (= 2 × 2 × 2). Check if that is true. Similarly,
from any point, the thickness after 10 folds increases by 1024 times
(= 2 multiplied by itself 10 times), as shown in the table below.
Fold Thickness Times increased by
0 to 10 1.024 cm – 0.001 cm 1.024 ÷ 0.001
= 1.023 cm = 1024
10 to 20 10.485 m – 1.024 cm 10.485 m ÷ 1.024 cm
≈ 10.474 m = 1024
20 to 30 10.737 km – 10.485 m 10.737 km ÷ 10.485 m
≈ 10.726 km = 1024
30 to 40 10995 km – 10.737 km 10995 km ÷
≈ 10984.2 km 10.737 km = 1024
2.2 Exponential Notation and Operations
The initial thickness of the paper was 0.001 cm.
Upon folding once, its thickness became 0.001 cm × 2 = 0.002 cm.
Folding it twice, its thickness became —
0.001 cm × 2 × 2 = 0.004 cm, or 0.001 cm × 22 = 0.004 cm (in shorthand).
Upon folding it thrice, its thickness became —
0.001 cm × 2 × 2 × 2, or 0.001 cm × 23 = 0.008 cm.
When folded four times, its thickness became —
0.001 cm × 2 × 2 × 2 × 2, or 0.001 cm × 24 = 0.016 cm.
Similarly, the expression for the thickness of the paper when folded 7
times will be 0.001 cm × 2 × 2 × 2 × 2 × 2 × 2 × 2, or 0.001 cm × 27 = 0.128 cm.
We have seen that square numbers can be expressed as n2 and cube
numbers as n3.
n × n = n2 (read as ‘n squared’ or ‘n raised to the power 2’)
21
Reprint 2026-27
, Ganita Prakash | Grade 8
n × n × n = n3 (read as ‘n cubed’ or ‘n raised to the power 3’)
n × n × n × n = n4 (read as ‘n raised to the power 4’ or ‘the 4th power of n’)
n × n × n × n × n × n × n = n7 (read as ‘n raised to the power 7’ or ‘the 7th
power of n’) and so on.
In general, we write na to denote n multiplied by itself a times.
54 = 5 × 5 × 5 × 5 = 625.
54 is the exponential form of 625. Here, 4 is the
exponent/power, and 5 is the base. Exponents of 5 is read as
4
the form 5n are called powers of 5: 51, 52, 53, 54, etc. ‘5 raised to the power 4’ or
‘5 to the power 4’ or
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 210 = 1024.
‘5 power 4’ or
Remember the 1024 from earlier? There, it meant ‘4th power of 5’
that after every 10 folds, the thickness increased
1024 times.
Which expression describes the thickness of a sheet of paper after
it is folded 10 times? The initial thickness is represented by the
letter-number v.
(i) 10v (ii) 10 + v (iii) 2 × 10 × v
(iv) 210 (v) 210v (vi) 102v
Some more examples of exponential notation:
4 × 4 × 4 = 43 = 64.
(– 4) × (– 4) × (– 4) = (– 4)3= – 64.
Similarly,
a × a × a × b × b can be expressed as a3b2 (read as a cubed b squared).
a × a × b × b × b × b can be expressed as a2b4 (read as a squared b raised
to the power 4).
Remember that 4 + 4 + 4 = 3 × 4 = 12, whereas 4 × 4 × 4 = 43 = 64.
Express the number 32400 as a product of its prime factors
2 32400
and represent the prime factors in their exponential form.
2 16200
32400 = 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 × 3 × 3. 2 8100
In exponential form, this would be 2 4050
32400 = 24 × 52 × 34. 5 2025
5 405
What is (– 1)5 ? Is it positive or negative? What about (– 1)56 ?
3 81
Is (– 2)4 = 16? Verify. What is 02, 05 ? 3 27
What is 0n ? 3 9
Figure it Out
3 3
1. Express the following in exponential form:
1
(i) 6×6×6×6 (ii) y×y
(iii) b×b×b×b (iv) 5×5×7×7×7
(v) 2×2×a×a (vi) a×a×a×c×c×c×c×d
22
Reprint 2026-27
