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Practice each question, then open the answer or worked solution.
Find the value of:
(i) 26
Solution:-
The above value can be written as,
= 2 × 2 × 2 × 2 × 2 × 2
= 64
(ii) 93
Solution:-
The above value can be written as,
= 9 × 9 × 9
= 729
(iii) 112
Solution:-
The above value can be written as,
= 11 × 11
= 121
(iv) 54
Solution:-
The above value can be written as,
= 5 × 5 × 5 × 5
= 625
Express the following in exponential form:
(i) 6 × 6 × 6 × 6
Solution:-
The given question can be expressed in the exponential form as 64.
(ii) t × t
Solution:-
The given question can be expressed in the exponential form as t2.
(iii) b × b × b × b
Solution:-
The given question can be expressed in the exponential form as b4.
(iv) 5 × 5× 7 × 7 × 7
Solution:-
The given question can be expressed in the exponential form as 52 × 73.
(v) 2 × 2 × a × a
Solution:-
The given question can be expressed in the exponential form as 22 × a2.
(vi) a × a × a × c × c × c × c × d
Solution:-
The given question can be expressed in the exponential form as a3 × c4 × d.
Express each of the following numbers using the exponential notation:
(i) 512
Solution:-
The factors of 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
So it can be expressed in the exponential form as 29.
(ii) 343
Solution:-
The factors of 343 = 7 × 7 × 7
So it can be expressed in the exponential form as 73.
(iii) 729
Solution:-
The factors of 729 = 3 × 3 × 3 × 3 × 3 × 3
So it can be expressed in the exponential form as 36.
(iv) 3125
Solution:-
The factors of 3125 = 5 × 5 × 5 × 5 × 5
So it can be expressed in the exponential form as 55.
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
So, 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
So, 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
So, 28 > 82
Hence, 28 is the greater number.
(iv) 1002 or 2100
Solution:-
The expansion of 1002 = 100 × 100 = 10000
The expansion of 2100
210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024
Then,
2100 = 1024 × 1024 ×1024 × 1024 ×1024 × 1024 × 1024 × 1024 × 1024 × 1024 = (1024)10
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.
Express each of the following as a 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
Simplify:
(i) 2 × 103
Solution:-
The above question can be written as,
= 2 × 10 × 10 × 10
= 2 × 1000
= 2000
(ii) 72 × 22
Solution:-
The above question can be written as,
= 7 × 7 × 2 × 2
= 49 × 4
= 196
(iii) 23 × 5
Solution:-
The above question can be written as,
= 2 × 2 × 2 × 5
= 8 × 5
= 40
(iv) 3 × 44
Solution:-
The above question can be written as,
= 3 × 4 × 4 × 4 × 4
= 3 × 256
= 768
(v) 0 × 102
Solution:-
The above question can be written as,
= 0 × 10 × 10
= 0 × 100
= 0
(vi) 52 × 33
Solution:-
The above question can be written as,
= 5 × 5 × 3 × 3 × 3
= 25 × 27
= 675
(vii) 24 × 32
Solution:-
The above question can be written as,
= 2 × 2 × 2 × 2 × 3 × 3
= 16 × 9
= 144
(viii) 32 × 104
Solution:-
The above question can be written as,
= 3 × 3 × 10 × 10 × 10 × 10
= 9 × 10000
= 90000
Simplify:
(i) (– 4)3
Solution:-
The expansion of -43
= – 4 × – 4 × – 4
= – 64
(ii) (–3) × (–2)3
Solution:-
The expansion of (-3) × (-2)3
= – 3 × – 2 × – 2 × – 2
= – 3 × – 8
= 24
(iii) (–3)2 × (–5)2
Solution:-
The expansion of (-3)2 × (-5)2
= – 3 × – 3 × – 5 × – 5
= 9 × 25
= 225
(iv) (–2)3 × (–10)3
Solution:-
The expansion of (-2)3 × (-10)3
= – 2 × – 2 × – 2 × – 10 × – 10 × – 10
= – 8 × – 1000
= 8000
Compare the following numbers:
(i) 2.7 × 1012 ; 1.5 × 108
Solution:-
By observing the question
Comparing the exponents of base 10,
Clearly,
2.7 × 1012 > 1.5 × 108
(ii) 4 × 1014 ; 3 × 1017
Solution:-
By observing the question
Comparing the exponents of base 10,
Clearly,
4 × 1014 < 3 × 1017