MATHEMATICS CLASS- 8
CHAPTER-2 (POWER PLAY)
Figure It Out-1
Chapter 2: Power Play
1. Express the following in exponential form:
(i) 6 × 6 × 6 × 6
Here, 6 is multiplied by itself 4 times.
Therefore, the exponential form is:
6⁴.
(ii) y × y
The variable y is multiplied by itself 2 times.
Therefore, the exponential form is:
y².
(iii) b × b × b × b
The variable b is multiplied by itself 4 times.
Therefore, the exponential form is:
b⁴.
(iv) 5 × 5 × 7 × 7 × 7
The factor 5 appears 2 times and the factor 7 appears 3 times.
Therefore, the exponential form is:
5² × 7³.
(v) 2 × 2 × a × a
The factor 2 appears 2 times and the variable a appears 2 times.
Therefore, the exponential form is:
2² × a².
(vi) a × a × a × c × c × c × c × d
The variable a appears 3 times, c appears 4 times and d appears once.
A factor occurring once is written without an exponent.
Therefore, the exponential form is:
a³ × c⁴ × d.
2. Express each of the following as a product of powers of their prime factors in exponential form.
(i) 648
Prime factorisation:
648 = 2 × 324
= 2 × 2 × 162
= 2 × 2 × 2 × 81
= 2 × 2 × 2 × 3 × 3 × 3 × 3
Therefore,
648 = 2³ × 3⁴.
(ii) 405
Prime factorisation:
405 = 5 × 81
= 5 × 3 × 3 × 3 × 3
Therefore,
405 = 3⁴ × 5.
(iii) 540
Prime factorisation:
540 = 54 × 10
= (2 × 3³) × (2 × 5)
= 2² × 3³ × 5
Therefore,
540 = 2² × 3³ × 5.
(iv) 3600
Prime factorisation:
3600 = 36 × 100
= (2² × 3²) × (2² × 5²)
Therefore,
3600 = 2⁴ × 3² × 5².
3. Write the numerical value of each of the following:
(i) 2 × 10³
10³ = 10 × 10 × 10 = 1000
Therefore,
2 × 10³ = 2 × 1000 = 2000.
(ii) 7² × 2³
7² = 49
2³ = 8
Therefore,
49 × 8 = 392.
(iii) 3 × 4⁴
4⁴ = 4 × 4 × 4 × 4 = 256
Therefore,
3 × 256 = 768.
(iv) (–3)² × (–5)²
(–3)² = 9
(–5)² = 25
Therefore,
9 × 25 = 225.
(v) 3² × 10⁴
3² = 9
10⁴ = 10000
Therefore,
9 × 10000 = 90000.
(vi) (–2)⁵ × (–10)⁶
(–2)⁵ = –32 (odd power gives a negative result)
(–10)⁶ = 1000000 (even power gives a positive result)
Therefore,
–32 × 1000000 = –32000000.
Figure It Out-2
1. Find out the units digit in the value of 2²²⁴ ÷ 4³². [Hint: 4 = 2²]
We have:
2²²⁴ ÷ 4³²
Since 4 = 2²,
4³² = (2²)³² = 2⁶⁴
Therefore,
2²²⁴ ÷ 2⁶⁴ = 2¹⁶⁰
Now we need the units digit of 2¹⁶⁰.
The units digits of powers of 2 repeat in a cycle:
2¹ = 2 → units digit 2
2² = 4 → units digit 4
2³ = 8 → units digit 8
2⁴ = 16 → units digit 6
After this, the pattern repeats every 4 powers.
160 ÷ 4 = 40 remainder 0
So the units digit will be the same as the units digit of 2⁴.
Answer: The units digit is 6.
2. There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would there be after 40 days?
Each container contains 5 bottles.
A new container is added every day.
Number of containers after 40 days = 40
Total bottles = 40 × 5
= 200
In scientific notation:
200 = 2 × 10²
Answer: 200 bottles (or 2 × 10² bottles).
3. Write the given number as the product of two or more powers in three different ways. The powers can be any integers.
(i) 64³
Since 64 = 2⁶,
64³ = (2⁶)³ = 2¹⁸
Three possible forms:
1. 2¹⁰ × 2⁸
2. 4⁵ × 4⁴
3. 8³ × 8³
(ii) 192⁸
Since 192 = 2⁶ × 3,
192⁸ = (2⁶ × 3)⁸
= 2⁴⁸ × 3⁸
Three possible forms:
1. 2⁴⁸ × 3⁸
2. 2⁴⁰ × 2⁸ × 3⁸
3. 2⁴⁰ × 6⁸
(iii) 32⁻⁵
Since 32 = 2⁵,
32⁻⁵ = (2⁵)⁻⁵
= 2⁻²⁵
Three possible forms:
1. 2⁻¹⁰ × 2⁻¹⁵
2. 2⁻⁵ × 2⁻²⁰
3. 4⁻¹² × 2⁻¹
4. Examine each statement below and find out if it is ‘Always True’, ‘Only Sometimes True’, or ‘Never True’. Explain your reasoning.
(i) Cube numbers are also square numbers.
A cube number is not always a square number.
For example:
8 = 2³ is not a square number.
However,
64 = 4³ = 8² is both a cube and a square.
Answer: Only Sometimes True.
(ii) Fourth powers are also square numbers.
Any fourth power can be written as:
n⁴ = (n²)²
Since it is the square of n², every fourth power is a square number.
Answer: Always True.
(iii) The fifth power of a number is divisible by the cube of that number.
n⁵ ÷ n³ = n²
Since the result is a whole-number power of n, n⁵ is divisible by n³.
Answer: Always True.
(iv) The product of two cube numbers is a cube number.
Let the cube numbers be a³ and b³.
a³ × b³ = (ab)³
The result is again a cube number.
Answer: Always True.
(v) q⁴⁶ is both a 4th power and a 6th power (q is a prime number).
For a number to be both a fourth power and a sixth power, its exponent must be divisible by both 4 and 6.
46 is not divisible by 4.
46 is not divisible by 6.
Therefore q⁴⁶ cannot be expressed as both a fourth power and a sixth power.
Answer: Never True.
5. Simplify and write these in the exponential form.
(i) 10⁻² × 10⁻⁵
Using the law:
am × an = am+n
10⁻² × 10⁻⁵ = 10⁻⁽²⁺⁵⁾
= 10⁻⁷
Answer: 10⁻⁷
(ii) 5⁷ ÷ 5⁴
Using the law:
am ÷ an = am−n
5⁷ ÷ 5⁴ = 5⁽⁷⁻⁴⁾
= 5³
Answer: 5³
(iii) 9⁻⁷ ÷ 9⁴
Using the law:
am ÷ an = am−n
9⁻⁷ ÷ 9⁴
= 9⁽⁻⁷⁻⁴⁾
= 9⁻¹¹
Answer: 9⁻¹¹
(iv) (13⁻²)⁻³
Using the law:
(am)n = amn
(13⁻²)⁻³
= 13⁽⁻²×⁻³⁾
= 13⁶
Answer: 13⁶
(v) m⁵n¹²(mn)⁹
First expand:
(mn)⁹ = m⁹n⁹
Therefore,
m⁵n¹² × m⁹n⁹
= m⁽⁵⁺⁹⁾ × n⁽¹²⁺⁹⁾
= m¹⁴n²¹
Answer: m¹⁴n²¹
6. If 12² = 144, what is:
(i) (1.2)²
1.2 = 12 ÷ 10
(1.2)² = 12² ÷ 10²
= 144 ÷ 100
= 1.44
Answer: 1.44
(ii) (0.12)²
0.12 = 12 ÷ 100
(0.12)² = 12² ÷ 100²
= 144 ÷ 10000
= 0.0144
Answer: 0.0144
(iii) (0.012)²
0.012 = 12 ÷ 1000
(0.012)² = 12² ÷ 1000²
= 144 ÷ 1000000
= 0.000144
Answer: 0.000144
(iv) 120²
120 = 12 × 10
120² = 12² × 10²
= 144 × 100
= 14400
Answer: 14400
7. Circle the numbers that are the same.
Given numbers:
2⁴ × 3⁶, 6⁴ × 3², 6¹⁰, 18² × 6², 6²⁴
2⁴ × 3⁶
= (2² × 3³)²
= 18²
6⁴ × 3²
= 2⁴ × 3⁶
18² × 6²
= (18 × 6)²
= 108²
Therefore the equal numbers are:
2⁴ × 3⁶ and 6⁴ × 3²
8. Identify the greater number in each of the following.
(i) 4³ or 3⁴
4³ = 64
3⁴ = 81
Greater number = 3⁴
(ii) 2⁸ or 8²
2⁸ = 256
8² = 64
Greater number = 2⁸
(iii) 100² or 2¹⁰⁰
100² = 10000
2¹⁰⁰ is an extremely large number.
Greater number = 2¹⁰⁰
9. A dairy plans to produce 8.5 billion packets of milk in a year. If the ID uses digits 0–9, how many digits should the code contain?
8.5 billion
= 8.5 × 10⁹
= 8,500,000,000
A code with n digits can form:
10ⁿ different codes.
10⁹ = 1 billion (not enough)
10¹⁰ = 10 billion (enough)
Answer: The code must contain 10 digits.
10. 64 is a square number (8²) and a cube number (4³). Are there other such numbers? Is there a way to describe such numbers in general?
Yes. Such numbers exist.
Examples:
1, 64, 729, 4096, ...
A number that is both a square and a cube is actually a sixth power because:
LCM(2,3) = 6
General form:
n⁶
Examples:
2⁶ = 64
3⁶ = 729
4⁶ = 4096
Answer: Every number of the form n⁶ is both a square and a cube.
11. A digital locker has an alphanumeric passcode of length 5. How many such codes are possible?
Alphanumeric means:
26 letters + 10 digits
= 36 characters
For each position there are 36 choices.
Total codes
= 36⁵
= 60,466,176
Answer: 36⁵ = 60,466,176 possible passcodes.
12. The worldwide population of sheep is about 10⁹ and that of goats is also about the same. What is the total population of sheep and goats?
Sheep population = 10⁹
Goat population = 10⁹
Total population
= 10⁹ + 10⁹
= 2 × 10⁹
Answer: 2 × 10⁹
13. Calculate and write the answer in scientific notation.
(i) 8 × 10⁹ people × 30 pieces
= 240 × 10⁹
= 2.4 × 10¹¹
Answer: 2.4 × 10¹¹ pieces
(ii) 100 million bee colonies × 50,000 bees
100 million = 10⁸
50,000 = 5 × 10⁴
Total bees
= 10⁸ × 5 × 10⁴
= 5 × 10¹²
Answer: 5 × 10¹² bees
(iii) 38 trillion bacteria × 8 billion humans
38 trillion = 3.8 × 10¹³
8 billion = 8 × 10⁹
Total bacteria
= (3.8 × 8) × 10²²
= 30.4 × 10²²
= 3.04 × 10²³
Answer: 3.04 × 10²³ bacteria
(iv) Total time spent eating in a lifetime in seconds.
This is an activity-based estimation question.
A reasonable estimate is:
≈ 1 × 10⁸ seconds
14. What was the date 1 arab / 1 billion seconds ago?
1 billion seconds
= 10⁹ seconds
10⁹ seconds ≈ 31.7 years
Counting back about 31.7 years from 2025–26 gives a date around:
1994
The exact date depends on the day from which the calculation is made.