Usting the identity `(a+ b)^(2) = (a^(2) + 2ab +b^(2)),` evaluate:
`(609)^(2)`
`(725)^(2)`
`(491)^(2)`
`(289)^(2)`

Answer

To evaluate the squares of the given numbers using the identity \((a + b)^2 = a^2 + 2ab + b^2\), we will follow these steps for each number.

### Step-by-Step Solutions:

1. **Evaluate \((609)^2\)**:
- Let \(a = 600\) and \(b = 9\).
- Using the identity:
\[ (609)^2 = (600 + 9)^2 = 600^2 + 2 \cdot 600 \cdot 9 + 9^2 \]
- Calculate each term:
- \(600^2 = 360000\)
- \(2 \cdot 600 \cdot 9 = 10800\)
- \(9^2 = 81\)
- Now, add these values:
\[ 360000 + 10800 + 81 = 370881 \]
- Therefore, \((609)^2 = 370881\).

2. **Evaluate \((725)^2\)**:
- Let \(a = 700\) and \(b = 25\).
- Using the identity:
\[ (725)^2 = (700 + 25)^2 = 700^2 + 2 \cdot 700 \cdot 25 + 25^2 \]
- Calculate each term:
- \(700^2 = 490000\)
- \(2 \cdot 700 \cdot 25 = 35000\)
- \(25^2 = 625\)
- Now, add these values:
\[ 490000 + 35000 + 625 = 525625 \]
- Therefore, \((725)^2 = 525625\).

3. **Evaluate \((491)^2\)**:
- Here, we can express \(491\) as \(500 - 9\).
- Let \(a = 500\) and \(b = 9\).
- Using the identity:
\[ (491)^2 = (500 - 9)^2 = 500^2 - 2 \cdot 500 \cdot 9 + 9^2 \]
- Calculate each term:
- \(500^2 = 250000\)
- \(-2 \cdot 500 \cdot 9 = -9000\)
- \(9^2 = 81\)
- Now, add these values:
\[ 250000 - 9000 + 81 = 241081 \]
- Therefore, \((491)^2 = 241081\).

4. **Evaluate \((289)^2\)**:
- Let \(a = 200\) and \(b = 89\).
- Using the identity:
\[ (289)^2 = (200 + 89)^2 = 200^2 + 2 \cdot 200 \cdot 89 + 89^2 \]
- Calculate each term:
- \(200^2 = 40000\)
- \(2 \cdot 200 \cdot 89 = 35600\)
- \(89^2 = 7921\)
- Now, add these values:
\[ 40000 + 35600 + 7921 = 83521 \]
- Therefore, \((289)^2 = 83521\).

### Final Results:
- \((609)^2 = 370881\)
- \((725)^2 = 525625\)
- \((491)^2 = 241081\)
- \((289)^2 = 83521\)