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Find the value of:
(i) 26 (ii) 93
(iii) 112 (iv)54
(i) 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64
(ii) 93 = 9 × 9 × 9 = 729
(iii) 112 = 11 × 11 = 121
(iv)54 = 5 × 5 × 5 × 5 = 625
Express the following in exponential form:
(i) 6 × 6 × 6 × 6 (ii) t × t
(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
(i) 6 × 6 × 6 × 6 = 64
(ii) t × t= t2
(iii) b × b × b × b = b4
(iv) 5 × 5 × 7 × 7 × 7 = 52 × 73
(v) 2 × 2 × a × a = 22 × a2
(vi) a × a × a × c × c × c × c × d = a3c4d
Express the following numbers using exponential notation:
(i) 512 (ii) 343
(iii) 729 (iv) 3125
(i) 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 29
(ii) 343 = 7 × 7 × 7 = 73
(iii) 729 = 3 × 3 × 3 × 3 × 3 × 3 = 36
(iv) 3125 = 5 × 5 × 5 × 5 × 5 = 55
Identify the greater number, wherever possible, in each of the following?
(i) 43 or 34 (ii) 53 or 35
(iii) 28 or 82 (iv) 1002 or 2100
(v) 210 or 102
(i) 43 = 4 × 4 × 4 = 64
34 = 3 × 3 × 3 × 3 = 81
Therefore, 34 > 43
(ii) 53 = 5 × 5 × 5 =125
35 = 3 × 3 × 3 × 3 × 3 = 243
Therefore, 35 > 53
(iii) 28 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
82 = 8 × 8 = 64
Therefore, 28 > 82
(iv)1002 or 2100
210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024
2100 = 1024 × 1024 × 1024 × 1024 × 1024 × 1024 × 1024 × 1024 ×1024 × 1024
1002 = 100 × 100 = 10000
Therefore, 2100 > 1002
(v) 210 and 102
210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024
102 = 10 × 10 = 100
Therefore, 210 > 102
Express each of the following as product of powers of their prime factors:
(i) 648 (ii) 405
(iii) 540 (iv) 3,600
(i) 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 = 23. 34
(ii) 405 = 3 × 3 × 3 × 3 × 5 = 34 . 5
(iii) 540 = 2 × 2 × 3 × 3 × 3 × 5 = 22. 33. 5
(iv) 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 = 24. 32. 52
Simplify:
(i) 2 × 103 (ii) 72 × 22
(iii) 23 × 5 (iv) 3× 44
(v) 0 × 102 (vi) 52 × 33
(vii) 24 × 32 (viii) 32 × 104
(i) 2 × 103 = 2 × 10 × 10 × 10 = 2 × 1000 = 2000
(ii) 72 × 22 = 7 × 7 × 2 × 2 = 49 × 4 = 196
(iii) 23 × 5 = 2 × 2 × 2 × 5 = 8 × 5 = 40
(iv) 3 × 44 = 3 × 4 × 4 × 4 × 4 = 3 × 256 = 768
(v) 0 × 102 = 0 × 10 × 10 = 0
(vi) 52 × 33 = 5 × 5 × 3 × 3 × 3 = 25 × 27 = 675
(vii) 24 × 32 = 2 × 2 × 2 × 2 × 3 × 3 = 16 × 9 = 144
(viii) 32 × 104 = 3 × 3 × 10 × 10 × 10 × 10 = 9 × 10000 = 90000
Simplify:
(i) (− 4)3 (ii) (− 3) × (− 2)3
(iii) (− 3)2 × (− 5)2 (iv)(− 2)3 × (−10)3
(i) (−4)3 = (−4) × (−4) × (−4) = −64
(ii) (−3) × (−2)3 = (−3) × (−2) × (−2) × (−2) = 24
(iii) (−3)2 × (−5)2 = (−3) × (−3) × (−5) × (−5) = 9 × 25 = 225
(iv) (−2)3 × (−10)3 = (−2) × (−2) × (−2) × (−10) × (−10) × (−10)
= (−8) × (−1000) = 8000
Compare the following numbers: (i) 2.7 × 1012; 1.5 × 108 (ii) 4 × 1014; 3 × 1017
(i) 2.7 × 1012; 1.5 × 108
2.7 × 1012 > 1.5 × 108
(ii) 4 × 1014; 3 × 1017
3 × 1017 > 4 × 1014
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