`sqrt (0.9)`=?

Options

A. `0.3`
B. `0.03`
C. `0.33`
D. `.94`

Answer

To find the square root of 0.9 using the division method, we can follow these steps:

### Step-by-Step Solution:

1. **Convert 0.9 to a Fraction**:
\[ 0.9 = \frac{9}{10} \]
This step helps us to work with whole numbers.

**Hint**: Converting decimals to fractions can simplify calculations.

2. **Set Up for the Division Method**:
We will treat 0.9 as 0.90, and we can represent it as 90.00 (adding pairs of zeros for the division method).

**Hint**: When using the division method, always pair the digits starting from the decimal point.

3. **Find the Largest Square**:
The largest square less than or equal to 90 is \(9^2 = 81\). Write down 9 above the square root line.

**Hint**: Look for the largest integer whose square is less than or equal to the number you are working with.

4. **Subtract and Bring Down**:
Subtract 81 from 90:
\[ 90 - 81 = 9 \]
Then bring down the next pair of zeros (00), making it 900.

**Hint**: Always subtract the square from the number and bring down the next pair of digits.

5. **Double the Current Quotient**:
The current quotient is 9. Double it to get 18.

**Hint**: Doubling the quotient helps in finding the next digit for the square root.

6. **Find the Next Digit**:
We need to find a digit \(x\) such that:
\[ (180 + x) \times x \leq 900 \]
Testing \(x = 4\):
\[ (180 + 4) \times 4 = 184 \times 4 = 736 \]
Testing \(x = 5\):
\[ (180 + 5) \times 5 = 185 \times 5 = 925 \quad (\text{too high}) \]
So, \(x = 4\) is the correct digit.

**Hint**: Test digits incrementally to find the largest possible value that satisfies the condition.

7. **Subtract Again**:
Subtract 736 from 900:
\[ 900 - 736 = 164 \]
Bring down another pair of zeros to make it 16400.

**Hint**: Continue the process of subtraction and bringing down pairs of zeros.

8. **Repeat the Process**:
Double the current quotient (94) to get 188. Now we need to find a digit \(y\) such that:
\[ (1880 + y) \times y \leq 16400 \]
Testing \(y = 8\):
\[ (1880 + 8) \times 8 = 1888 \times 8 = 15104 \]
Testing \(y = 9\):
\[ (1880 + 9) \times 9 = 1889 \times 9 = 17001 \quad (\text{too high}) \]
So, \(y = 8\) is the correct digit.

**Hint**: Continue testing until you find the largest digit that fits.

9. **Final Calculation**:
After the second iteration, we have:
\[ \text{Square root of } 0.9 \approx 0.948 \]

**Hint**: Keep track of the digits you find to get a more accurate approximation.

### Final Answer:
\[ \sqrt{0.9} \approx 0.948 \]