Evaluate the following products without multiplying directly:

(i) 103 × 107

(ii) 95 × 96

(iii) 104 × 96

(i) 103 × 107

We can write it as,

= (100 + 3)(100 + 7)

[∵ (x + a)(x + b) = x² + (a + b)x + ab]

Putting x = 100, a = 3, b = 7

= (100)² + (3 + 7)(100) + (3)(7)

= 10000 + (10)(100) + 21

= 10000 + 1000 + 21

= 11021

Hence, 103 × 107 = 11021

(ii) 95 × 96

We can write it as,

[100 + (-5)][100 + (-4)]

[∵ (x + a)(x + b) = x² + (a + b)x + ab]

Putting x = 100, a = -5 , b = -4

= (100)² - [(-5) + (-4)](100) + (-5) x (-4)

= 10000 - (9)(100) + 20

= 10000 - 900 + 20

= 10020 - 900

= 9120

Hence, 95 × 96 = 9120

(iii) 104 × 96

We can write it as,

(100 + 4)(100 - 4)

[∵ (a + b)(a - b) = a² - b²]

Putting a = 100, b = 4

= (100)² - (4)²

= 10000 - 16

= 9984

Hence, 104 × 96 = 9984