To evaluate the square roots of the given numbers, we will use the long division method. Let's go through each part step by step.
### Step 1: Evaluate `sqrt(42.25)`
1. **Pair the digits**: Start from the decimal point and pair the digits. Here, we have (42)(25).
2. **Find the largest square**: The largest square less than or equal to 42 is 36 (which is \(6^2\)). So, we write down 6.
3. **Subtract**: \(42 - 36 = 6\).
4. **Bring down the next pair**: Bring down 25 to get 625.
5. **Double the quotient**: Double 6 to get 12. Now, we need to find a digit \(x\) such that \(12x \cdot x \leq 625\). Testing \(x = 5\), we find \(125 \cdot 5 = 625\).
6. **Subtract**: \(625 - 625 = 0\).
7. **Result**: Therefore, \( \sqrt{42.25} = 6.5\).
### Step 2: Evaluate `sqrt(1.96)`
1. **Pair the digits**: Pair the digits as (1)(96).
2. **Find the largest square**: The largest square less than or equal to 1 is 1 (which is \(1^2\)). So, we write down 1.
3. **Subtract**: \(1 - 1 = 0\).
4. **Bring down the next pair**: Bring down 96 to get 96.
5. **Double the quotient**: Double 1 to get 2. Now, we need to find a digit \(x\) such that \(2x \cdot x \leq 96\). Testing \(x = 4\), we find \(24 \cdot 4 = 96\).
6. **Subtract**: \(96 - 96 = 0\).
7. **Result**: Therefore, \( \sqrt{1.96} = 1.4\).
### Step 3: Evaluate `sqrt(6.4009)`
1. **Pair the digits**: Pair the digits as (6)(40)(09).
2. **Find the largest square**: The largest square less than or equal to 6 is 4 (which is \(2^2\)). So, we write down 2.
3. **Subtract**: \(6 - 4 = 2\).
4. **Bring down the next pair**: Bring down 40 to get 240.
5. **Double the quotient**: Double 2 to get 4. Now, we need to find a digit \(x\) such that \(4x \cdot x \leq 240\). Testing \(x = 5\), we find \(45 \cdot 5 = 225\).
6. **Subtract**: \(240 - 225 = 15\).
7. **Bring down the next pair**: Bring down 09 to get 1509.
8. **Double the quotient**: Double 25 to get 50. Now, we need to find a digit \(x\) such that \(50x \cdot x \leq 1509\). Testing \(x = 3\), we find \(53 \cdot 3 = 159\) which is too high. Testing \(x = 2\), we find \(52 \cdot 2 = 104\).
9. **Subtract**: \(1509 - 104 = 1405\).
10. **Result**: Therefore, \( \sqrt{6.4009} \approx 2.53\).
### Step 4: Evaluate `sqrt(0.4225)`
1. **Pair the digits**: Pair the digits as (0)(42)(25).
2. **Find the largest square**: The largest square less than or equal to 0 is 0. So, we write down 0.
3. **Bring down the next pair**: Bring down 42 to get 42.
4. **Find the largest square**: The largest square less than or equal to 42 is 36 (which is \(6^2\)). So, we write down 6.
5. **Subtract**: \(42 - 36 = 6\).
6. **Bring down the next pair**: Bring down 25 to get 625.
7. **Double the quotient**: Double 6 to get 12. Now, we need to find a digit \(x\) such that \(12x \cdot x \leq 625\). Testing \(x = 5\), we find \(125 \cdot 5 = 625\).
8. **Subtract**: \(625 - 625 = 0\).
9. **Result**: Therefore, \( \sqrt{0.4225} = 0.65\).
### Step 5: Evaluate `sqrt(2)`
1. **Use long division method**: Start with 2. Pair it as (2)(00).
2. **Find the largest square**: The largest square less than or equal to 2 is 1 (which is \(1^2\)). So, we write down 1.
3. **Subtract**: \(2 - 1 = 1\).
4. **Bring down the next pair**: Bring down 00 to get 100.
5. **Double the quotient**: Double 1 to get 2. Now, we need to find a digit \(x\) such that \(2x \cdot x \leq 100\). Testing \(x = 4\), we find \(24 \cdot 4 = 96\).
6. **Subtract**: \(100 - 96 = 4\).
7. **Bring down the next pair**: Bring down 00 to get 400.
8. **Double the quotient**: Double 14 to get 28. Now, we need to find a digit \(x\) such that \(28x \cdot x \leq 400\). Testing \(x = 1\), we find \(281 \cdot 1 = 281\).
9. **Subtract**: \(400 - 281 = 119\).
10. **Result**: Therefore, \( \sqrt{2} \approx 1.41\).
### Step 6: Evaluate `sqrt(0.8)`
1. **Pair the digits**: Pair the digits as (0)(80).
2. **Find the largest square**: The largest square less than or equal to 0 is 0. So, we write down 0.
3. **Bring down the next pair**: Bring down 80 to get 80.
4. **Find the largest square**: The largest square less than or equal to 80 is 64 (which is \(8^2\)). So, we write down 8.
5. **Subtract**: \(80 - 64 = 16\).
6. **Bring down the next pair**: Bring down 00 to get 1600.
7. **Double the quotient**: Double 8 to get 16. Now, we need to find a digit \(x\) such that \(16x \cdot x \leq 1600\). Testing \(x = 9\), we find \(169 \cdot 9 = 1521\).
8. **Subtract**: \(1600 - 1521 = 79\).
9. **Result**: Therefore, \( \sqrt{0.8} \approx 0.89\).
### Summary of Results:
- \( \sqrt{42.25} = 6.5 \)
- \( \sqrt{1.96} = 1.4 \)
- \( \sqrt{6.4009} \approx 2.53 \)
- \( \sqrt{0.4225} = 0.65 \)
- \( \sqrt{2} \approx 1.41 \)
- \( \sqrt{0.8} \approx 0.89 \)
