Let's solve this step-by-step.

We are given the equation:

$4 \cdot 365 \cdot 20 = \frac{2\left(0.005^{\frac{-1.3}{2}} - x^{\frac{-1.3}{2}}\right)}{\left(-1.3\right) \cdot 0.24 \cdot 10^{-11} \cdot 1.5^{3.3} \cdot 25^{3.3} \cdot \pi^{1.65}}$

Step 1: Simplify the left-hand side

The left-hand side is:

$4 \cdot 365 \cdot 20$

First, multiply:

$4 \cdot 365 = 1460$

Then:

$1460 \cdot 20 = 29200$

So, the left-hand side is 29200.

Step 2: Simplify the denominator of the right-hand side

Let's break down the denominator:

$\left(-1.3\right) \cdot 0.24 \cdot 10^{-11} \cdot 1.5^{3.3} \cdot 25^{3.3} \cdot \pi^{1.65}$

We can simplify each part individually.

• $1.5^{3.3} \approx 3.4205$
• $25^{3.3} \approx 3280.627$
• $\pi^{1.65} \approx 3.6735$

Now, multiplying all the constants together:

$-1.3 \cdot 0.24 \cdot 10^{-11} \cdot 3.4205 \cdot 3280.627 \cdot 3.6735$

This simplifies to:

$-1.3 \cdot 0.24 \cdot 10^{-11} \cdot 41810.737$

$-1.3 \cdot 0.24 = -0.312$

Now:

$-0.312 \cdot 10^{-11} \cdot 41810.737 = -1.303 \cdot 10^{-7}$

So the denominator simplifies to approximately:

$-1.303 \cdot 10^{-7}$

Step 3: Rearrange the equation

We now have:

$29200 = \frac{2\left(0.005^{\frac{-1.3}{2}} - x^{\frac{-1.3}{2}}\right)}{-1.303 \cdot 10^{-7}}$

Multiply both sides by $-1.303 \cdot 10^{-7}$:

$29200 \cdot (-1.303 \cdot 10^{-7}) = 2\left(0.005^{\frac{-1.3}{2}} - x^{\frac{-1.3}{2}}\right)$

$-3.8046 \cdot 10^{-3} = 2\left(0.005^{\frac{-1.3}{2}} - x^{\frac{-1.3}{2}}\right)$

Step 4: Simplify further

Now, divide by 2:

$-1.9023 \cdot 10^{-3} = 0.005^{\frac{-1.3}{2}} - x^{\frac{-1.3}{2}}$

So, solving for $x^{\frac{-1.3}{2}}$:

$x^{\frac{-1.3}{2}} = 0.005^{\frac{-1.3}{2}} + 1.9023 \cdot 10^{-3}$

Next, calculate $0.005^{\frac{-1.3}{2}}$:

$0.005^{\frac{-1.3}{2}} \approx 55.081$

So:

$x^{\frac{-1.3}{2}} = 55.081 + 1.9023 \cdot 10^{-3}$

$x^{\frac{-1.3}{2}} \approx 55.083$

Step 5: Solve for $x$

Now, raise both sides to the power of $\frac{2}{-1.3}$:

$x = \left(55.083\right)^{\frac{2}{-1.3}}$

This gives:

$x \approx 2.110$

Thus, the solution for $x$ is approximately:

$x \approx 2.110$

---

Would you like more details or explanations? Feel free to ask!

Here are some related questions to explore further:

1. How do negative exponents affect calculations, particularly when solving for a variable?
2. What is the significance of scientific notation in equations like this one?
3. Can this method be applied to other equations involving negative exponents?
4. How do you handle complex denominators like the one in this equation?
5. What are the practical applications of equations similar to the one above?

Tip: When working with exponents, especially negative ones, remember that they indicate inverse powers, and it’s crucial to maintain precision when dealing with such small or large values.