Let's break down the given expression step by step. The original expression is:

$\frac{\left[1 + 96\sqrt{2 \cdot 0.285(1 - 0.285)} + 0.84\sqrt{0.24(1 - 0.2) + 0.41(1 - 0.36)}\right]^2}{(0.24 - 0.41)^2}$

Step 1: Simplify the square root terms

First square root term:

$96\sqrt{2 \cdot 0.285(1 - 0.285)} = 96\sqrt{2 \cdot 0.285 \cdot 0.715}$ $2 \cdot 0.285 \cdot 0.715 = 0.4071$ $96\sqrt{0.4071} \approx 96 \cdot 0.638 \approx 61.25$

Second square root term:

$0.84\sqrt{0.24(1 - 0.2) + 0.41(1 - 0.36)} = 0.84\sqrt{0.24 \cdot 0.8 + 0.41 \cdot 0.64}$ $0.24 \cdot 0.8 = 0.192, \quad 0.41 \cdot 0.64 = 0.2624$ $0.192 + 0.2624 = 0.4544$ $0.84\sqrt{0.4544} \approx 0.84 \cdot 0.674 = 0.566$

Step 2: Simplify the numerator

Now, substitute these results into the numerator: $1 + 61.25 + 0.566 = 62.816$ $\left(62.816\right)^2 = 3945.688$

Step 3: Simplify the denominator

$(0.24 - 0.41)^2 = (-0.17)^2 = 0.0289$

Step 4: Final computation

Now, divide the numerator by the denominator: $\frac{3945.688}{0.0289} \approx 136524.93$

Final Answer:

$\boxed{136524.93}$

Would you like more details or have any questions?

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Here are 5 related questions to explore further:

1. What is the general form of the binomial theorem?
2. How do square roots affect the simplification of algebraic expressions?
3. How can irrational numbers like square roots be approximated?
4. What are some common methods to solve non-linear equations?
5. How can you simplify expressions involving multiple square roots?

Tip: When dealing with square roots in complex expressions, simplify the terms step by step to avoid errors.