Evaluate:
`sqrt (17956)`

Answer

To evaluate the square root of 17956, we will use the long division method. Here’s a step-by-step solution:

### Step 1: Pair the Digits
Start by pairing the digits of the number from right to left. For 17956, we can pair it as (17)(95)(56).

### Step 2: Find the Largest Square
Find the largest square number less than or equal to the first pair (17). The largest square is \(4^2 = 16\).

### Step 3: Subtract and Bring Down
Subtract 16 from 17, which gives us 1. Now, bring down the next pair (95) next to it, making it 195.

### Step 4: Double the Quotient
The quotient so far is 4. Double it to get 8. We will now find a digit \(x\) such that \(8x \times x\) is less than or equal to 195.

### Step 5: Find the Next Digit
Testing \(x = 2\):
- \(82 \times 2 = 164\) (which is less than 195)
Testing \(x = 3\):
- \(83 \times 3 = 249\) (which is greater than 195)

So, we take \(x = 2\).

### Step 6: Subtract Again
Now, subtract 164 from 195:
- \(195 - 164 = 31\)
Now, bring down the next pair (56) to make it 3156.

### Step 7: Double the New Quotient
The current quotient is 42 (from 4 and 2). Double it to get 84. We will now find a digit \(y\) such that \(84y \times y\) is less than or equal to 3156.

### Step 8: Find the Next Digit
Testing \(y = 3\):
- \(843 \times 3 = 2529\) (which is less than 3156)
Testing \(y = 4\):
- \(844 \times 4 = 3376\) (which is greater than 3156)

So, we take \(y = 3\).

### Step 9: Final Subtraction
Now, subtract 2529 from 3156:
- \(3156 - 2529 = 627\)
Now, bring down the next pair (00) to make it 62700.

### Step 10: Continue the Process
Double the current quotient (423) to get 846. We will find a digit \(z\) such that \(846z \times z\) is less than or equal to 62700.

Testing \(z = 7\):
- \(8467 \times 7 = 59269\) (which is less than 62700)
Testing \(z = 8\):
- \(8468 \times 8 = 67744\) (which is greater than 62700)

So, we take \(z = 7\).

### Final Calculation
Now, subtract \(59269\) from \(62700\):
- \(62700 - 59269 = 3431\)

At this point, we have found that the square root of 17956 is approximately \(134\) since we have completed the long division.

### Conclusion
Thus, \( \sqrt{17956} = 134 \).

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