Find the square root of 147.1369.

Answer

To find the square root of 147.1369 using the long division method, follow these steps:

### Step 1: Prepare the Number
Start by writing the number 147.1369 in pairs, starting from the decimal point. This gives us:
- 1 47 . 13 69

### Step 2: Find the Largest Square
Look for the largest square number less than or equal to the first pair (1). The largest square is 1 (since \(1^2 = 1\)). Write 1 above the line.

### Step 3: Subtract and Bring Down
Subtract 1 from 1, which gives us 0. Bring down the next pair (47), making it 047.

### Step 4: Double the Quotient
Double the quotient (1), which gives us 2. Now, we need to find a digit \(x\) such that \(2x \cdot x\) is less than or equal to 47.

### Step 5: Find the Next Digit
Testing \(x = 2\):
- \(2 \cdot 2 = 4\)
- \(24 \cdot 2 = 48\) (too high)
Testing \(x = 1\):
- \(2 \cdot 1 = 2\)
- \(21 \cdot 1 = 21\) (valid)

So, we take \(x = 2\). Write 2 above the line next to 1, making it 12.

### Step 6: Subtract Again
Now, \(21\) from \(47\) gives us \(26\). Bring down the next pair (13), making it 2613.

### Step 7: Double the Quotient Again
Double the current quotient (12), giving us 24. Now, we need to find a digit \(y\) such that \(24y \cdot y\) is less than or equal to 2613.

### Step 8: Find the Next Digit
Testing \(y = 1\):
- \(241 \cdot 1 = 241\) (valid)
Testing \(y = 2\):
- \(242 \cdot 2 = 484\) (valid)
Testing \(y = 3\):
- \(243 \cdot 3 = 729\) (valid)
Testing \(y = 4\):
- \(244 \cdot 4 = 976\) (valid)
Testing \(y = 5\):
- \(245 \cdot 5 = 1225\) (valid)
Testing \(y = 6\):
- \(246 \cdot 6 = 1476\) (valid)
Testing \(y = 7\):
- \(247 \cdot 7 = 1729\) (too high)

So, we take \(y = 6\). Write 6 above the line, making it 12.6.

### Step 9: Subtract Again
Now, \(1476\) from \(2613\) gives us \(1137\). Bring down the next pair (69), making it 113769.

### Step 10: Double the Quotient Again
Double the current quotient (126), giving us 252. Now, we need to find a digit \(z\) such that \(252z \cdot z\) is less than or equal to 113769.

### Step 11: Find the Next Digit
Testing \(z = 4\):
- \(2524 \cdot 4 = 10096\) (valid)
Testing \(z = 5\):
- \(2525 \cdot 5 = 12625\) (too high)

So, we take \(z = 4\). Write 4 above the line, making it 12.13.

### Step 12: Final Subtraction
Now, \(10096\) from \(113769\) gives us \(12773\). Since we have reached the desired precision, we can conclude.

### Final Answer
The square root of 147.1369 is **12.13**.

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