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Evaluate the following products without multiplying directly: (i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96

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Evaluate the following products without multiplying directly:

(i) 103 × 107

(ii) 95 × 96

(iii) 104 × 96

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(i) 103×107

103 × 107= (100 + 3) × (100 + 7)

Using identity, [(x + a)(x + b) = x2+(a + b)x + ab

Here, x = 100

a = 3

b = 7

We get, 103×107 = (100+3)×(100+7)

= (100)2+(3+7)100+(3×7))

= 10000+1000+21

= 11021

(ii) 95×96  

95 × 96 = (100-5)×(100-4)

Using identity, [(x-a)(x-b) = x2-(a+b)x+ab

Here, x = 100

a = -5

b = -4

We get, 95×96 = (100-5)×(100-4)

= (100)2+100(-5+(-4))+(-5×-4)

= 10000-900+20

= 9120

(iii) 104×96

104×96 = (100+4)×(100–4)

Using identity, [(a+b)(a-b)= a2-b2]

Here, a = 100

b = 4

We get, 104×96 = (100+4)×(100–4)

= (100)2–(4)2

= 10000–16

= 9984

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