Evaluate:
`sqrt (19600)`

Answer

To evaluate \( \sqrt{19600} \), we can use the long division method for finding square roots. Here’s a step-by-step solution:

### Step 1: Pair the digits
Start from the right and pair the digits of the number. For 19600, we can pair them as follows:
- (19)(60)(00)

### Step 2: Find the largest square
Now, we look at the first pair (19). The largest perfect square less than or equal to 19 is 16, which is \( 4^2 \).

### Step 3: Subtract and bring down the next pair
Subtract 16 from 19:
\[ 19 - 16 = 3 \]
Now, bring down the next pair (60) to get 360.

### 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 \( (80 + x)x \) is less than or equal to 360.

### Step 5: Find the appropriate digit
We test \( x = 4 \):
\[ (80 + 4) \times 4 = 84 \times 4 = 336 \]
This is less than 360. Now, we try \( x = 5 \):
\[ (80 + 5) \times 5 = 85 \times 5 = 425 \]
This is greater than 360. So, we take \( x = 4 \).

### Step 6: Subtract and bring down the next pair
Now, subtract 336 from 360:
\[ 360 - 336 = 24 \]
Bring down the next pair (00) to get 2400.

### Step 7: Double the quotient
Now, our quotient is 44. Double it to get 88. We will find a digit \( y \) such that \( (880 + y)y \) is less than or equal to 2400.

### Step 8: Find the appropriate digit
Testing \( y = 2 \):
\[ (880 + 2) \times 2 = 882 \times 2 = 1764 \]
This is less than 2400. Now, we try \( y = 3 \):
\[ (880 + 3) \times 3 = 883 \times 3 = 2649 \]
This is greater than 2400. So, we take \( y = 2 \).

### Step 9: Final subtraction
Now, subtract 1764 from 2400:
\[ 2400 - 1764 = 636 \]
Since we have completed the process, we can conclude that the square root of 19600 is:
\[ \sqrt{19600} = 140 \]

### Final Answer
Thus, \( \sqrt{19600} = 140 \).

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