The expression in the image appears to be a complex formula. Let's break down and simplify it:

$\frac{0.5 \times 20}{415} \left[ 1 - \sqrt{1 - \frac{4.6 \times 27.36 \times 10^6}{20 \times 1000 \times 205 \times 205}} \right] \times 1000 \times 205$

Step-by-Step Solution:

1. Calculate $\frac{0.5 \times 20}{415}$: $\frac{0.5 \times 20}{415} = \frac{10}{415} \approx 0.0241$

2. Evaluate the term inside the square root: $1 - \frac{4.6 \times 27.36 \times 10^6}{20 \times 1000 \times 205 \times 205}$

   Let's calculate each part:

   • Numerator: $4.6 \times 27.36 \times 10^6 = 125.856 \times 10^6 = 125856000$
   • Denominator: $20 \times 1000 \times 205 \times 205 = 841000000$

   So, the term becomes: $1 - \frac{125856000}{841000000}$

3. Simplify the fraction: $\frac{125856000}{841000000} \approx 0.1497$

   Therefore: $1 - 0.1497 = 0.8503$

4. Take the square root: $\sqrt{0.8503} \approx 0.9221$

5. Subtract from 1: $1 - 0.9221 = 0.0779$

6. Multiply all the parts together: $0.0241 \times 0.0779 \times 1000 \times 205$

   Step-by-step:

   • $0.0241 \times 0.0779 \approx 0.001878$
   • $0.001878 \times 1000 = 1.878$
   • $1.878 \times 205 \approx 385.99$

Final Answer:

$\boxed{385.99}$

Would you like further details or have any questions?

Related Questions:

1. How do you simplify square roots of non-perfect squares?
2. What is the process to calculate fractions with large numbers efficiently?
3. Can you explain how the order of operations (PEMDAS) affects complex expressions?
4. How can you check your results when dealing with approximations?
5. What are some common strategies for simplifying expressions with multiple terms?

Tip:

Always simplify expressions step-by-step to minimize errors, especially when dealing with large or complex calculations.