Using the formula for squaring a binomial, evaluate the following:
(i) `(54)^(2)` (ii) `(82)^(2)` (iii) `(103)^(2)` (iv) `(704)^(2)`

Answer

To solve the given question using the formula for squaring a binomial, we will apply the identity:

\[ (a + b)^2 = a^2 + b^2 + 2ab \]

where \(a\) and \(b\) are parts of the binomial.

### (i) Evaluate \( (54)^2 \)

1. **Identify \(a\) and \(b\)**:
- Let \(a = 50\) and \(b = 4\) (since \(54 = 50 + 4\)).

2. **Apply the formula**:
\[ (54)^2 = (50 + 4)^2 = 50^2 + 4^2 + 2 \cdot 50 \cdot 4 \]

3. **Calculate each term**:
- \(50^2 = 2500\)
- \(4^2 = 16\)
- \(2 \cdot 50 \cdot 4 = 400\)

4. **Combine the results**:
\[ (54)^2 = 2500 + 16 + 400 = 2916 \]

### (ii) Evaluate \( (82)^2 \)

1. **Identify \(a\) and \(b\)**:
- Let \(a = 80\) and \(b = 2\) (since \(82 = 80 + 2\)).

2. **Apply the formula**:
\[ (82)^2 = (80 + 2)^2 = 80^2 + 2^2 + 2 \cdot 80 \cdot 2 \]

3. **Calculate each term**:
- \(80^2 = 6400\)
- \(2^2 = 4\)
- \(2 \cdot 80 \cdot 2 = 320\)

4. **Combine the results**:
\[ (82)^2 = 6400 + 4 + 320 = 6724 \]

### (iii) Evaluate \( (103)^2 \)

1. **Identify \(a\) and \(b\)**:
- Let \(a = 100\) and \(b = 3\) (since \(103 = 100 + 3\)).

2. **Apply the formula**:
\[ (103)^2 = (100 + 3)^2 = 100^2 + 3^2 + 2 \cdot 100 \cdot 3 \]

3. **Calculate each term**:
- \(100^2 = 10000\)
- \(3^2 = 9\)
- \(2 \cdot 100 \cdot 3 = 600\)

4. **Combine the results**:
\[ (103)^2 = 10000 + 9 + 600 = 10609 \]

### (iv) Evaluate \( (704)^2 \)

1. **Identify \(a\) and \(b\)**:
- Let \(a = 700\) and \(b = 4\) (since \(704 = 700 + 4\)).

2. **Apply the formula**:
\[ (704)^2 = (700 + 4)^2 = 700^2 + 4^2 + 2 \cdot 700 \cdot 4 \]

3. **Calculate each term**:
- \(700^2 = 490000\)
- \(4^2 = 16\)
- \(2 \cdot 700 \cdot 4 = 5600\)

4. **Combine the results**:
\[ (704)^2 = 490000 + 16 + 5600 = 495616 \]

### Final Results:
- (i) \( (54)^2 = 2916 \)
- (ii) \( (82)^2 = 6724 \)
- (iii) \( (103)^2 = 10609 \)
- (iv) \( (704)^2 = 495616 \)