https://file.aiquickdraw.com/mathbot/file/172994911892919vkvj6x.jpg

The expression in the image is as follows:

\[
A_{st} = \frac{\frac{0.5 \times 20}{415}}{1 - \sqrt{\frac{4.6 \times 19.28 \times 10^6}{20 \times 1000 \times 150}}}
\]

Let's break down and solve each part step-by-step.

1. **Top Part of the Numerator**:
   \[
   0.5 \times 20 = 10
   \]
   
   So, the numerator simplifies to:
   \[
   \frac{10}{415} = 0.0241 \text{ (approximately)}
   \]

2. **Inner Part of the Denominator's Square Root**:
   \[
   4.6 \times 19.28 \times 10^6 = 885.68 \times 10^6 = 8.8568 \times 10^8
   \]

   Now, for the denominator of this inner term:
   \[
   20 \times 1000 \times 150 = 3000000
   \]

   Thus, the entire term inside the square root becomes:
   \[
   \frac{8.8568 \times 10^8}{3000000} = 295.2267
   \]

3. **Square Root Calculation**:
   \[
   \sqrt{295.2267} \approx 17.179
   \]

4. **Final Denominator Calculation**:
   \[
   1 - 17.179 = -16.179
   \]

5. **Complete Expression**:
   \[
   A_{st} = \frac{0.0241}{-16.179} \approx -0.00149
   \]

So, the final answer is approximately:

\[
A_{st} \approx -0.00149
\]

Would you like a more detailed breakdown of any part?

### Related Questions:

1. How does taking the square root of a negative number affect the expression?
2. What are possible interpretations if the final result is negative?
3. How could variations in constants (like 0.5 or 20) affect the outcome?
4. How does each part contribute to the overall calculation in engineering or physics contexts?
5. What are common mistakes when evaluating nested fractions and square roots?

### Tip:
For complex expressions, break down the calculation into smaller, manageable steps to minimize errors.

Calculate A_st from the given expression involving nested fractions and a square root.

Learn how to simplify and solve a nested fraction expression involving square roots. This detailed step-by-step solution covers algebraic simplification, square root operations, and proper fraction handling. Suitable for advanced high school students.

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