Using the formula for squaring a binomial evaluate the following:
(i) `(69)^(2)` (ii) `(78)^(2)` (iii) `(197)^(2)` (iv) `(999)^(2)`

Answer

To evaluate the squares of the given numbers using the formula for squaring a binomial, we will use the identity:

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

Now, let's solve each part step by step.

### (i) Evaluating \( (69)^2 \)

1. **Rewrite 69**: We can express 69 as \( 70 - 1 \).
2. **Apply the formula**: Using the identity, we have:
\[ (70 - 1)^2 = 70^2 + 1^2 - 2 \cdot 70 \cdot 1 \]
3. **Calculate each term**:
- \( 70^2 = 4900 \)
- \( 1^2 = 1 \)
- \( 2 \cdot 70 \cdot 1 = 140 \)
4. **Combine the results**:
\[ 4900 + 1 - 140 = 4901 - 140 = 4761 \]

**Final answer for (i)**: \( 4761 \)

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### (ii) Evaluating \( (78)^2 \)

1. **Rewrite 78**: We can express 78 as \( 80 - 2 \).
2. **Apply the formula**: Using the identity, we have:
\[ (80 - 2)^2 = 80^2 + 2^2 - 2 \cdot 80 \cdot 2 \]
3. **Calculate each term**:
- \( 80^2 = 6400 \)
- \( 2^2 = 4 \)
- \( 2 \cdot 80 \cdot 2 = 320 \)
4. **Combine the results**:
\[ 6400 + 4 - 320 = 6404 - 320 = 6084 \]

**Final answer for (ii)**: \( 6084 \)

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### (iii) Evaluating \( (197)^2 \)

1. **Rewrite 197**: We can express 197 as \( 200 - 3 \).
2. **Apply the formula**: Using the identity, we have:
\[ (200 - 3)^2 = 200^2 + 3^2 - 2 \cdot 200 \cdot 3 \]
3. **Calculate each term**:
- \( 200^2 = 40000 \)
- \( 3^2 = 9 \)
- \( 2 \cdot 200 \cdot 3 = 1200 \)
4. **Combine the results**:
\[ 40000 + 9 - 1200 = 40009 - 1200 = 38809 \]

**Final answer for (iii)**: \( 38809 \)

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### (iv) Evaluating \( (999)^2 \)

1. **Rewrite 999**: We can express 999 as \( 1000 - 1 \).
2. **Apply the formula**: Using the identity, we have:
\[ (1000 - 1)^2 = 1000^2 + 1^2 - 2 \cdot 1000 \cdot 1 \]
3. **Calculate each term**:
- \( 1000^2 = 1000000 \)
- \( 1^2 = 1 \)
- \( 2 \cdot 1000 \cdot 1 = 2000 \)
4. **Combine the results**:
\[ 1000000 + 1 - 2000 = 1000001 - 2000 = 998001 \]

**Final answer for (iv)**: \( 998001 \)

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### Summary of Answers:
- (i) \( 4761 \)
- (ii) \( 6084 \)
- (iii) \( 38809 \)
- (iv) \( 998001 \)