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Using laws of exponents, simplify and write the answer in exponential form:
(i) 32 × 34 × 38 (ii) 615 ÷ 610 (iii) a3 × a2
(iv) 7x× 72 (v) 
(vi) 25 × 55
(vii) a4 × b4 (viii) (34)3
(ix) 
(x) 8t ÷ 82
(i) 32 × 34 × 38 = (3)2 + 4 + 8 (am × an = am+n)
= 314
(ii) 615 ÷ 610 = (6)15 − 10 (am ÷ an = am−n)
= 65
(iii) a3 × a2 = a(3 + 2) (am × an = am+n)
= a5
(iv) 7x + 72 = 7x + 2 (am × an = am+n)
(v) (52)3 ÷ 53
= 52 × 3 ÷ 53 (am)n = amn
= 56 ÷ 53
= 5(6 − 3) (am ÷ an = am−n)
= 53
(vi) 25 × 55
= (2 × 5)5 [am × bm = (a × b)m]
= 105
(vii) a4 × b4
= (ab)4 [am × bm = (a × b)m]
(viii) (34)3 = 34 × 3 = 312 (am)n = amn
(ix) (220 ÷ 215) × 23
= (220 − 15)× 23 (am ÷ an = am−n)
= 25 × 23
= (25 + 3) (am × an = am+n)
= 28
(x) 8t ÷ 82 = 8(t − 2) (am ÷ an = am−n)
Simplify and express each of the following in exponential form:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 20 + 30 + 40
(vii) 20 × 30 × 40 (viii) (30 + 20) × 50 (ix) 
(x) 
(xi) 
(xii) 
(i)
(ii) [(52)3 × 54] ÷ 57
= [52 × 3 × 54] ÷ 57 (am)n = amn
= [56 × 54] ÷ 57
= [56 + 4] ÷ 57 (am × an = am+n)
= 510 ÷ 57
= 510 − 7 (am ÷ an = am−n)
= 53
(iii) 254 ÷ 53 = (5 ×5)4 ÷ 53
= (52)4 ÷ 53
= 52 × 4 ÷ 53 (am)n = amn
= 58 ÷ 53
= 58 − 3 (am ÷ an = am−n)
= 55
(iv)
= 1 × 7 × 115 = 7 × 115
(v)
(vi) 20 + 30 + 40 = 1 + 1 + 1 = 3
(vii) 20 × 30 × 40 = 1 × 1 × 1 = 1
(viii) (30 + 20) × 50 = (1 + 1) × 1 = 2
(ix)
(x)
(xi)
(xii) (23 × 2)2 = 
(am × an = am+n)
= (24)2 = 24 × 2 (am)n = amn
= 28
Say true or false and justify your answer:
(i) 10 × 1011 = 10011 (ii) 23 > 52
(iii) 23 × 32 = 65 (iv) 30 = (1000)0
(i) 10 × 1011 = 10011
L.H.S. = 10 × 1011 = 1011 + 1 (am × an = am+n)
= 1012
R.H.S. = 10011 = (10 ×10)11= (102)11
= 102 × 11 = 1022 (am)n = amn
As L.H.S. ≠ R.H.S.,
Therefore, the given statement is false.
(ii) 23 > 52
L.H.S. = 23 = 2 × 2 × 2 = 8
R.H.S. = 52 = 5 × 5 = 25
As 25 > 8,
Therefore, the given statement is false.
(iii) 23 × 32 = 65
L.H.S. = 23 × 32 = 2 × 2 × 2 × 3 × 3 = 72
R.H.S. = 65 = 7776
As L.H.S. ≠ R.H.S.,
Therefore, the given statement is false.
(iv) 30 = (1000)0
L.H.S. = 30 = 1
R.H.S. = (1000)0 = 1 = L.H.S.
Therefore, the given statement is true.
Express each of the following as a product of prime factors only in exponential form:
(i) 108 × 192 (ii) 270
(iii) 729 × 64(iv) 768
(i) 108 × 192
= (2 × 2 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2 × 3)
= (22 × 33) × (26 × 3)
= 26 + 2 × 33 + 1 (am × an = am+n)
= 28 × 34
(ii) 270 = 2 × 3 × 3 × 3 × 5 = 2 × 33 × 5
(iii) 729 × 64 = (3 × 3 × 3 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2)
= 36 × 26
(iv) 768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 = 28 × 3
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(iii) 
