Find the value of `sqrt(37249)`.

Options

A. 197
B. 187
C. 183
D. 193

Answer

To find the value of \( \sqrt{37249} \) using the long division method, follow these steps:

### Step 1: Pair the digits
Start by pairing the digits of the number from right to left. For \( 37249 \), we can pair it as follows:
- The last pair is \( 49 \)
- The next pair is \( 72 \)
- The remaining digit is \( 3 \)

So, we have:
- \( 3 \) (alone)
- \( 72 \) (pair)
- \( 49 \) (pair)

### Step 2: Find the largest square
Now, we start with the leftmost pair (which is \( 3 \)). The largest square less than or equal to \( 3 \) is \( 1^2 = 1 \).

Write \( 1 \) above the \( 3 \) and subtract \( 1 \) from \( 3 \):
\[ 3 - 1 = 2 \]

### Step 3: Bring down the next pair
Next, bring down the pair \( 72 \) next to \( 2 \), making it \( 272 \).

### Step 4: Double the quotient
Double the quotient we have so far (which is \( 1 \)):
\[ 2 \times 1 = 2 \]
Now we need to find a digit \( x \) such that \( (20 + x) \times x \) is less than or equal to \( 272 \).

Testing \( x = 9 \):
\[ (20 + 9) \times 9 = 29 \times 9 = 261 \]
This is less than \( 272 \).

### Step 5: Subtract and bring down the next pair
Now, write \( 9 \) above the \( 272 \) and subtract \( 261 \):
\[ 272 - 261 = 11 \]
Now bring down the next pair \( 49 \) to make it \( 1149 \).

### Step 6: Double the new quotient
Now, double the quotient (which is now \( 19 \)):
\[ 2 \times 19 = 38 \]
We need to find a digit \( y \) such that \( (380 + y) \times y \) is less than or equal to \( 1149 \).

Testing \( y = 3 \):
\[ (380 + 3) \times 3 = 383 \times 3 = 1149 \]
This is exactly equal to \( 1149 \).

### Step 7: Write the final quotient
Now, write \( 3 \) above the \( 1149 \) and subtract:
\[ 1149 - 1149 = 0 \]

The final quotient is \( 193 \), which means:
\[ \sqrt{37249} = 193 \]

### Final Answer
Thus, the value of \( \sqrt{37249} \) is \( 193 \).

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