The positive square root of 45.5625 is

Options

A. 5.25
B. 5.65
C. 6.35
D. 6.75

Answer

To find the positive square root of 45.5625, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Number**: We need to find the positive square root of 45.5625.

2. **Pair the Digits**: Start from the decimal point and pair the digits. The number can be grouped as (45)(56)(25).

3. **Find the Largest Square**: We need to find the largest square number less than or equal to 45. The perfect squares less than 45 are:
- 1² = 1
- 2² = 4
- 3² = 9
- 4² = 16
- 5² = 25
- 6² = 36
- 7² = 49 (too large)

The largest square is 6² = 36.

4. **Subtract and Bring Down**: Subtract 36 from 45:
\[ 45 - 36 = 9 \]
Now, bring down the next pair (56) to make it 956.

5. **Double the Root**: The current root is 6. Double it to get 12. We will now find a digit (let's call it 'x') such that:
\[ (12x) \times x \leq 956 \]

6. **Test Values for x**:
- If x = 7:
\[ 127 \times 7 = 889 \quad (\text{valid}) \]
- If x = 8:
\[ 128 \times 8 = 1024 \quad (\text{too large}) \]
So, we take x = 7.

7. **Subtract Again**: Now subtract 889 from 956:
\[ 956 - 889 = 67 \]
Bring down the next pair (25) to make it 6725.

8. **Double the Current Root**: The current root is 67. Double it to get 134. Now, find a digit (let's call it 'y') such that:
\[ (134y) \times y \leq 6725 \]

9. **Test Values for y**:
- If y = 5:
\[ 1345 \times 5 = 6725 \quad (\text{valid}) \]
- If y = 6:
\[ 1346 \times 6 = 8076 \quad (\text{too large}) \]
So, we take y = 5.

10. **Final Calculation**: Now we have found both digits:
- The square root of 45.5625 is 67.5.

### Final Answer:
The positive square root of 45.5625 is **6.75**.