NCERT Solutions for Class 8 Maths – Ganita Prakash
Chapter 2: Power Play
Exercise 2.2
1. Find out the units digit in the value of 2224 ÷ 432. [Hint: 4 = 22]
Given:
2224 ÷ 432
Using 4 = 22, we get:
432 = (22)32 = 264
So,
2224 ÷ 432 = 2224 ÷ 264
Using the law of exponents, am ÷ an = am-n,
2224 ÷ 264 = 2160
Now, the units digit of powers of 2 follows the pattern:
2, 4, 8, 6
This pattern repeats after every 4 powers.
160 ÷ 4 leaves remainder 0
So, the units digit will be the same as the 4th term of the pattern, that is 6.
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 be there after 40 days?
Each container has 5 bottles.
Every day, 1 new container is brought.
So, after 40 days, number of containers = 40
Total number of bottles = 40 × 5 = 200
Answer: 200 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
We know 64 = 26.
Three different ways are:
643 = 26 × 26 × 26
643 = 46 × 43
643 = 84 × 82
(ii) 1928
Three different ways are:
1928 = 1923 × 1923 × 1922
1928 = 1925 × 1923
1928 = 1924 × 1922 × 1922
(iii) 32−5
Three different ways are:
32−5 = 32−2 × 32−3
32−5 = 2−10 × 2−15
32−5 = 4−3 × 4−2 × 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 number can be both a cube and a square only when it is a perfect sixth power.
Example: 64 = 82 and also 43
So, this is Only Sometimes True.
(ii) Fourth powers are also square numbers.
a4 = (a2)2
So every fourth power is a square number.
This is Always True.
(iii) The fifth power of a number is divisible by the cube of that number.
a5 ÷ a3 = a2
This is true for all non-zero numbers.
So, this is Always True.
(iv) The product of two cube numbers is a cube number.
Let the two cube numbers be a3 and b3.
Their product = a3 × b3 = (ab)3
So, the product is again a cube number.
This is Always True.
(v) q46 is both a 4th power and a 6th power (q is a prime number).
For a number to be a 4th power, the exponent must be divisible by 4.
46 is not divisible by 4.
For a number to be a 6th power, the exponent must be divisible by 6.
46 is not divisible by 6.
So, q46 is neither a 4th power nor a 6th power.
This is Never True.
5. Simplify and write these in the exponential form.
(i) 10−2 × 10−5
Using am × an = am+n,
10−2 × 10−5 = 10−7
Answer: 10−7
(ii) 57 ÷ 54
Using am ÷ an = am−n,
57 ÷ 54 = 53
Answer: 53
(iii) 9−7 ÷ 94
9−7 ÷ 94 = 9−7−4 = 9−11
Answer: 9−11
(iv) (13−2)−3
Using (am)n = amn,
(13−2)−3 = 136
Answer: 136
(v) m5n12(mn)9
First, expand:
(mn)9 = m9n9
So,
m5n12(mn)9 = m5n12 × m9n9
= m14n21
Answer: m14n21
6. If 122 = 144, what is
(i) (1.2)2
1.2 = 12 ÷ 10
(1.2)2 = (12 ÷ 10)2 = 122 ÷ 102
= 144 ÷ 100 = 1.44
Answer: 1.44
(ii) (0.12)2
0.12 = 12 ÷ 100
(0.12)2 = (12 ÷ 100)2 = 122 ÷ 1002
= 144 ÷ 10000 = 0.0144
Answer: 0.0144
(iii) (0.012)2
0.012 = 12 ÷ 1000
(0.012)2 = (12 ÷ 1000)2 = 122 ÷ 10002
= 144 ÷ 1000000 = 0.000144
Answer: 0.000144
(iv) 1202
120 = 12 × 10
1202 = (12 × 10)2 = 122 × 102
= 144 × 100 = 14400
Answer: 14400
NCERT Solutions for Class 8 Maths – Ganita Prakash
Chapter 2: Power Play
Exercise 2.2 (Remaining Questions)
7. Circle the numbers that are the same:
Given:
24 × 36, 64 × 32, 610, 182 × 62, 624
Convert all into prime factor form:
24 × 36 = 24 × 36
64 × 32 = (2×3)4 × 32
= 24 × 34 × 32 = 24 × 36
182 × 62 = (2×32)2 × (2×3)2
= 22 × 34 × 22 × 32
= 24 × 36
610 = 210 × 310 (different)
624 = 224 × 324 (different)
Same numbers are:
24 × 36, 64 × 32, 182 × 62
8. Identify the greater number:
(i) 43 or 34
43 = 64
34 = 81
Greater: 34
(ii) 28 or 82
28 = 256
82 = 64
Greater: 28
(iii) 1002 or 2100
1002 = 10000
2100 is a very large number (greater than 1030)
Greater: 2100
9. A dairy plans to produce 8.5 billion packets of milk in a year. How many digits should the code consist of?
8.5 billion = 8.5 × 109
We need a code system using digits 0–9.
Number of possible codes with n digits = 10n
We need 10n ≥ 8.5 × 109
109 = 1 billion (not enough)
1010 = 10 billion (sufficient)
Answer: The code should consist of 10 digits.
10. Are there numbers that are both squares and cubes?
Yes. Such numbers are perfect sixth powers.
Example: 64 = 82 = 43 = 26
In general, numbers of the form a6 are both perfect squares and cubes.
11. How many alphanumeric passcodes of length 5 are possible?
Each position can have 26 letters + 10 digits = 36 choices.
Total number of codes = 365
Answer: 365 possible codes
12. Total population of sheep and goats:
Sheep = 109
Goats = 109
Total = 109 + 109 = 2 × 109
Answer: 2 × 109
13. Write the answers in scientific notation:
(i)
Assume world population ≈ 1010
Total clothes = 30 × 1010 = 3 × 1011
Answer: 3 × 1011
(ii)
100 million = 108
Total bees = 108 × 5 × 104 = 5 × 1012
Answer: 5 × 1012
(iii)
38 trillion = 3.8 × 1013
Humans ≈ 1010
Total bacteria = 3.8 × 1013 × 1010
= 3.8 × 1023
Answer: 3.8 × 1023
(iv)
Approximate eating time in life ≈ 10 years
Seconds in 1 year ≈ 3.15 × 107
Total ≈ 10 × 3.15 × 107 = 3.15 × 108
Answer: ≈ 3.15 × 108 seconds
14. What was the date 1 billion seconds ago?
1 billion seconds = 109 seconds
Convert to years:
109 ÷ (3.15 × 107) ≈ 31.7 years
So, approximately 32 years ago from today.
Answer: About 32 years ago