Given that `sqrt(574.6) = 23.97`
`sqrt(5746) = 75.8` then `sqrt(0.00005746)` equals

Options

A. `0.002397`
B. `0.0002397`
C. `0.007580`
D. `0.00758`

Answer

To find the value of \( \sqrt{0.00005746} \), we can use the properties of square roots and the values given in the problem.

### Step-by-Step Solution:

1. **Rewrite the number in scientific notation**:
\[ 0.00005746 = 5.746 \times 10^{-5} \]

2. **Apply the square root property**:
\[ \sqrt{0.00005746} = \sqrt{5.746 \times 10^{-5}} = \sqrt{5.746} \times \sqrt{10^{-5}} \]

3. **Calculate \( \sqrt{10^{-5}} \)**:
\[ \sqrt{10^{-5}} = 10^{-5/2} = 10^{-2.5} = \frac{1}{10^{2.5}} = \frac{1}{\sqrt{100} \times 10} = \frac{1}{10 \times 10} = \frac{1}{31.622} \approx 0.031622 \]

4. **Use the given value of \( \sqrt{574.6} \)**:
From the problem, we know:
\[ \sqrt{574.6} \approx 23.97 \]
To find \( \sqrt{5.746} \), we can relate it to \( \sqrt{574.6} \):
\[ \sqrt{5.746} = \sqrt{\frac{574.6}{100}} = \frac{\sqrt{574.6}}{\sqrt{100}} = \frac{23.97}{10} \approx 2.397 \]

5. **Combine the results**:
Now, we combine the results from steps 3 and 4:
\[ \sqrt{0.00005746} = \sqrt{5.746} \times \sqrt{10^{-5}} \approx 2.397 \times 0.031622 \]

6. **Calculate the final value**:
\[ 2.397 \times 0.031622 \approx 0.0758 \]

### Final Answer:
\[ \sqrt{0.00005746} \approx 0.0758 \]