1. Find the value of:
(i) 26
Solution:
We can write the given value as:
= 2 × 2 × 2 × 2 × 2 × 2
= 64
(ii) 93
Solution:
We can write the above value as:
= 9 × 9 × 9
= 729
(iii) 112
Solution:
We can write the above value as:
= 11 × 11
= 121
(iv) 54
Solution:
We can write the given value as:
= 5 × 5 × 5 × 5
= 625
2. Express the following in exponential form:
(i) 6 × 6 × 6 × 6
Solution:
We can express the given question in the exponential form of 64.
(ii) t × t
Solution:
We can express the given question in the exponential form as t2.
(iii) b × b × b × b
Solution:
We can express the given question in the exponential form as b4.
(iv) 5 × 5× 7 × 7 × 7
Solution:
We can express the given question in the exponential form as 52 × 73.
(v) 2 × 2 × a × a
Solution:
We can express the given question in the exponential form as 22 × a2.
(vi) a × a × a × c × c × c × c × d
Solution:
We can express the given question in the exponential form as a3 × c4 × d.
3. Express each of the following numbers using exponential notation:
(i) 512
Solution:
The factors of 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
Hence, it can be expressed in the exponential form as 29.
(ii) 343
Solution:
The factors of 343 = 7 × 7 × 7
Hence, it can be expressed in the exponential form as 73.
(iii) 729
Solution:
The factors of 729 = 3 × 3 × 3 × 3 × 3 × 3
Hence, it can be expressed in the exponential form as 36.
(iv) 3125
Solution:
The factors of 3125 = 5 × 5 × 5 × 5 × 5
Hence, it can be expressed in the exponential form as 55.
4. Identify the greater number, wherever possible, in each of the following?
(i) 43 or 34
Solution:
The expansion of 43 = 4 × 4 × 4 = 64
The expansion of 34 = 3 × 3 × 3 × 3 = 81
Clearly,
64 < 81
Then, 43 < 34
Hence 34 is the greater number.
(ii) 53 or 35
Solution:
The expansion of 53 = 5 × 5 × 5 = 125
The expansion of 35 = 3 × 3 × 3 × 3 × 3= 243
Clearly,
125 < 243
Then, 53 < 35
Hence, 35 is the greater number.
(iii) 28 or 82
Solution:
The expansion of 28 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
The expansion of 82 = 8 × 8= 64
Clearly,
256 > 64
Then, 28 > 82
Hence, 28 is the greater number.
(iv) 1002 or 2100
Solution:
The expansion of 1002 = 100 × 100 = 10000
The expansion of 214
214 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 =16384
Clearly,
1002 < 2100
Hence, 2100 is the greater number.
(v) 210 or 102
Solution:
The expansion of 210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024
The expansion of 102 = 10 × 10= 100
Clearly,
1024 > 100
So, 210 > 102
Hence 210 is the greater number.
5. Express each of the following as product of powers of their prime factors:
(i) 648
Solution:
Factors of 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3
= 23 × 34
(ii) 405
Solution:
Factors of 405 = 3 × 3 × 3 × 3 × 5
= 34 × 5
(iii) 540
Solution:
Factors of 540 = 2 × 2 × 3 × 3 × 3 × 5
= 22 × 33 × 5
(iv) 3,600
Solution:
Factors of 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5
= 24 × 32 × 52
6. Simplify:
(i) 2 × 103
Solution:
= 2 × 10 × 10 × 10
= 2 × 1000
= 2000
(ii) 72 × 22
Solution:
= 7 × 7 × 2 × 2
= 49 × 4
= 196
(iii) 23 × 5
Solution:
= 2 × 2 × 2 × 5
= 8 × 5
= 40
(iv) 3 × 44
Solution:
= 3 × 4 × 4 × 4 × 4
= 3 × 256
= 768
(v) 0 × 102
Solution:
= 0 × 10 × 10
= 0
(vi) 52 × 33
Solution:
= 5 × 5 × 3 × 3 × 3
= 25 × 27
= 675
(vii) 24 × 32
Solution:
= 2 × 2 × 2 × 2 × 3 × 3
= 16 × 9
= 144
(viii) 32 × 104
Solution:
= 3 × 3 × 10 × 10 × 10 × 10
= 9 × 10000
= 90000
7. Simplify:
(i) (– 4) 3
Solution:
= – 4 × – 4 × – 4
= – 64
(ii) (–3) × (–2) 3
Solution:
= – 3 × – 2 × – 2 × – 2
= – 3 × – 8
= 24
(iii) (–3) 2 × (–5) 2
Solution:
= – 3 × – 3 × – 5 × – 5
= 9 × 25
= 225
(iv) (–2) 3 × (–10) 3
Solution:
= – 2 × – 2 × – 2 × – 10 × – 10 × – 10
= – 8 × – 1000
= 8000
8. Compare the following numbers:
(i) 2.7 × 1012 ; 1.5 × 108
Solution:
By comparing the exponents of base 10,
Clearly,
2.7 × 1012 > 1.5 × 108
(ii) 4 × 1014 ; 3 × 1017
Solution:
By comparing the exponents of base 10,
Clearly,
4 × 1014 < 3 × 1017
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