Evaluate the square root of 22.09 by long division method.

Options

A. 4.7
B. 4.5
C. 5.4
D. 5.2

Answer

To evaluate the square root of 22.09 using the long division method, follow these steps:

### Step-by-Step Solution:

1. **Pair the Digits**:
Start from the decimal point and pair the digits. For 22.09, we have:
- Pair 1: 22
- Pair 2: 09

2. **Find the Largest Square**:
Determine the largest integer whose square is less than or equal to the first pair (22).
- The squares of integers are:
- \(4^2 = 16\)
- \(5^2 = 25\)
- The largest square less than 22 is \(4^2 = 16\).

3. **Subtract and Bring Down**:
Subtract 16 from 22:
\[ 22 - 16 = 6 \]
Now, bring down the next pair (09), making it 609.

4. **Double the Quotient**:
Double the quotient obtained so far (which is 4):
\[ 2 \times 4 = 8 \]

5. **Find the Next Digit**:
We need to find a digit \(d\) such that:
\[ (80 + d) \times d \leq 609 \]
- Testing \(d = 7\):
\[ 87 \times 7 = 609 \]
This works perfectly.

6. **Final Calculation**:
Subtract \(609\) from \(609\):
\[ 609 - 609 = 0 \]
The quotient is \(4.7\).

Thus, the square root of 22.09 is:
\[ \sqrt{22.09} = 4.7 \]