Let's break down the given equation step-by-step:

$\left( \frac{(0.032)^5}{(0.2 \times L_e)} \right)^{\frac{1}{2}} = \left( \frac{(0.06)^5}{(0.2 \times 600)} \right)^{\frac{1}{2}} + \left( \frac{(0.02^9)^5}{(0.02 \times 600)} \right)^{\frac{1}{2}}$

Step 1: Simplify each term inside the square root.

Left-hand side:

$\left( \frac{(0.032)^5}{(0.2 \times L_e)} \right)^{\frac{1}{2}}$ This is a square root of the fraction, so we first simplify $(0.032)^5$.

$(0.032)^5 = 3.3554432 \times 10^{-7}$

Thus, the expression becomes:

$\left( \frac{3.3554432 \times 10^{-7}}{(0.2 \times L_e)} \right)^{\frac{1}{2}}$

Right-hand side:

Let's handle the two terms separately.

1. First term:

$\left( \frac{(0.06)^5}{(0.2 \times 600)} \right)^{\frac{1}{2}}$ Simplify $(0.06)^5$:

$(0.06)^5 = 7.776 \times 10^{-7}$

So, the term becomes:

$\left( \frac{7.776 \times 10^{-7}}{0.2 \times 600} \right)^{\frac{1}{2}} = \left( \frac{7.776 \times 10^{-7}}{120} \right)^{\frac{1}{2}}$

2. Second term:

$\left( \frac{(0.02^9)^5}{(0.02 \times 600)} \right)^{\frac{1}{2}}$ Simplify $(0.02^9)^5$:

$(0.02^9)^5 = (1.024 \times 10^{-12})^5 \quad (\text{which will be a very small number})$

Thus, this term will contribute a very small value. We will calculate this explicitly if needed.

Step 2: Solve for $L_e$

To isolate $L_e$, you will need to numerically evaluate both sides of the equation and solve for $L_e$. Would you like to proceed with the numerical evaluation, or do you need further clarification on any part?

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Here are 5 related questions you might find useful:

1. How can I use logarithms to solve equations involving exponents and square roots?
2. How do I handle very small numbers in scientific notation?
3. What is the physical interpretation of $L_e$ in this equation?
4. How would the result change if $L_e$ was increased or decreased by a factor of 10?
5. How do I convert between different units in equations involving physical constants?

Tip: When solving equations with square roots or exponents, always handle the terms step by step and check for simplifications at each stage.