Evaluate the following products without multiplying directly :
(i) `103xx107`
(ii) `95xx96`
(iii) `104xx96`

Answer

To evaluate the products without direct multiplication, we can use algebraic identities. Let's solve each product step by step.

### (i) Evaluate \( 103 \times 107 \)

1. **Identify the form**: We can express \( 103 \) and \( 107 \) in terms of \( 100 \):
\[ 103 = 100 + 3 \quad \text{and} \quad 107 = 100 + 7 \]

2. **Use the identity**: We can use the identity \( (x + a)(x + b) = x^2 + (a + b)x + ab \). Here, \( x = 100 \), \( a = 3 \), and \( b = 7 \).

3. **Substitute into the identity**:
\[ = 100^2 + (3 + 7) \times 100 + (3 \times 7) \]

4. **Calculate each term**:
- \( 100^2 = 10000 \)
- \( (3 + 7) \times 100 = 10 \times 100 = 1000 \)
- \( 3 \times 7 = 21 \)

5. **Combine the results**:
\[ 10000 + 1000 + 21 = 11021 \]

Thus, \( 103 \times 107 = 11021 \).

### (ii) Evaluate \( 95 \times 96 \)

1. **Identify the form**: We can express \( 95 \) and \( 96 \) in terms of \( 100 \):
\[ 95 = 100 - 5 \quad \text{and} \quad 96 = 100 - 4 \]

2. **Use the identity**: We can use the same identity \( (x + a)(x + b) = x^2 + (a + b)x + ab \). Here, \( x = 100 \), \( a = -5 \), and \( b = -4 \).

3. **Substitute into the identity**:
\[ = 100^2 + (-5 - 4) \times 100 + (-5) \times (-4) \]

4. **Calculate each term**:
- \( 100^2 = 10000 \)
- \( (-5 - 4) \times 100 = -9 \times 100 = -900 \)
- \( (-5) \times (-4) = 20 \)

5. **Combine the results**:
\[ 10000 - 900 + 20 = 9120 \]

Thus, \( 95 \times 96 = 9120 \).

### (iii) Evaluate \( 104 \times 96 \)

1. **Identify the form**: We can express \( 104 \) and \( 96 \) in terms of \( 100 \):
\[ 104 = 100 + 4 \quad \text{and} \quad 96 = 100 - 4 \]

2. **Use the identity**: This is a special case where we can use the difference of squares identity \( (a + b)(a - b) = a^2 - b^2 \). Here, \( a = 100 \) and \( b = 4 \).

3. **Substitute into the identity**:
\[ = 100^2 - 4^2 \]

4. **Calculate each term**:
- \( 100^2 = 10000 \)
- \( 4^2 = 16 \)

5. **Combine the results**:
\[ 10000 - 16 = 9984 \]

Thus, \( 104 \times 96 = 9984 \).

### Summary of Results:
- \( 103 \times 107 = 11021 \)
- \( 95 \times 96 = 9120 \)
- \( 104 \times 96 = 9984 \)