Evaluate the following products without multiplying directly:

(i) 103×107

(ii) 95×96

(iii) 104×96

Let's evaluate each of the products without directly multiplying:

(i) 103 × 107:

We can use the identity for the difference of squares:
(a + b)(a - b) = a² - b².

Here, let’s rewrite the expression:
103 × 107 = (105 - 2) × (105 + 2)

Now, applying the difference of squares formula: = 105² - 2²
= 11025 - 4
= 11021

So, 103 × 107 = 11021.

(ii) 95 × 96:

We can apply a similar strategy by rewriting the numbers around 95 and 96. Notice that 95 and 96 are close to 100. We can use the identity:
(a - b)(a + b) = a² - b².

Here, 95 = 100 - 5 and 96 = 100 - 4, so we can use:
95 × 96 = (100 - 5) × (100 - 4)

This becomes: = 100² - (5 + 4) × 100 + 5 × 4
= 10000 - 900 + 20
= 9100 + 20
= 9120

So, 95 × 96 = 9120.

(iii) 104 × 96:

Again, we can apply the same technique using numbers close to 100. Let’s rewrite the expression:
104 × 96 = (100 + 4) × (100 - 4)

This is a difference of squares: = 100² - 4²
= 10000 - 16
= 9984

So, 104 × 96 = 9984.

Thus, the results are:

(i) 103 × 107 = 11021
(ii) 95 × 96 = 9120
(iii) 104 × 96 = 9984