Using the formula `(a+b)^(2) = (a^(2) + 2ab+b ^(2)),` evaluate:
`(310)^(2)`
`(508)^(2)`
`(630)^(2)`
`(196)^(2)`
`(689)^(2)`

Answer

To evaluate the squares of the given numbers using the formula \((a+b)^2 = a^2 + 2ab + b^2\), we will break each number into two parts \(a\) and \(b\) and apply the formula step by step.

### Step-by-Step Solutions:

1. **Evaluate \( (310)^2 \)**:
- Break down: \( 310 = 300 + 10 \)
- Here, \( a = 300 \) and \( b = 10 \).
- Calculate:
- \( a^2 = 300^2 = 90000 \)
- \( 2ab = 2 \times 300 \times 10 = 6000 \)
- \( b^2 = 10^2 = 100 \)
- Combine:
\[ (310)^2 = a^2 + 2ab + b^2 = 90000 + 6000 + 100 = 96100 \]

2. **Evaluate \( (508)^2 \)**:
- Break down: \( 508 = 500 + 8 \)
- Here, \( a = 500 \) and \( b = 8 \).
- Calculate:
- \( a^2 = 500^2 = 250000 \)
- \( 2ab = 2 \times 500 \times 8 = 8000 \)
- \( b^2 = 8^2 = 64 \)
- Combine:
\[ (508)^2 = a^2 + 2ab + b^2 = 250000 + 8000 + 64 = 258064 \]

3. **Evaluate \( (630)^2 \)**:
- Break down: \( 630 = 600 + 30 \)
- Here, \( a = 600 \) and \( b = 30 \).
- Calculate:
- \( a^2 = 600^2 = 360000 \)
- \( 2ab = 2 \times 600 \times 30 = 36000 \)
- \( b^2 = 30^2 = 900 \)
- Combine:
\[ (630)^2 = a^2 + 2ab + b^2 = 360000 + 36000 + 900 = 396900 \]

4. **Evaluate \( (196)^2 \)**:
- Break down: \( 196 = 190 + 6 \)
- Here, \( a = 190 \) and \( b = 6 \).
- Calculate:
- \( a^2 = 190^2 = 36100 \)
- \( 2ab = 2 \times 190 \times 6 = 2280 \)
- \( b^2 = 6^2 = 36 \)
- Combine:
\[ (196)^2 = a^2 + 2ab + b^2 = 36100 + 2280 + 36 = 38416 \]

5. **Evaluate \( (689)^2 \)**:
- Break down: \( 689 = 600 + 89 \)
- Here, \( a = 600 \) and \( b = 89 \).
- Calculate:
- \( a^2 = 600^2 = 360000 \)
- \( 2ab = 2 \times 600 \times 89 = 106800 \)
- \( b^2 = 89^2 = 7921 \)
- Combine:
\[ (689)^2 = a^2 + 2ab + b^2 = 360000 + 106800 + 7921 = 474721 \]

### Final Results:
- \( (310)^2 = 96100 \)
- \( (508)^2 = 258064 \)
- \( (630)^2 = 396900 \)
- \( (196)^2 = 38416 \)
- \( (689)^2 = 474721 \)