Let's break down the equation you provided and solve it step by step:

The equation is:

$\left(\frac{(0.032)^5}{0.2 \times L_e}\right)^{1/2} = \left(\frac{(0.06)^5}{0.2 \times 600}\right)^{1/2} + \left(\frac{(0.029)^5}{0.02 \times 600}\right)^{1/2}$

Step 1: Simplify the right-hand side

Start by simplifying the two terms on the right-hand side separately.

• For the first term: $\frac{(0.06)^5}{0.2 \times 600}$ First, calculate $0.06^5$: $0.06^5 = 0.00007776$ Now calculate $0.2 \times 600 = 120$. Thus, the first term becomes: $\frac{0.00007776}{120} = 0.000000648$ Now, take the square root: $\sqrt{0.000000648} \approx 0.000804$

• For the second term: $\frac{(0.029)^5}{0.02 \times 600}$ First, calculate $0.029^5$: $0.029^5 = 0.000000205379$ Now calculate $0.02 \times 600 = 12$. Thus, the second term becomes: $\frac{0.000000205379}{12} = 0.000000017113$ Now, take the square root: $\sqrt{0.000000017113} \approx 0.000131$

Step 2: Add the two terms on the right-hand side

Now add the results from the two terms: $0.000804 + 0.000131 = 0.000935$

Step 3: Simplify the left-hand side

Now, let's simplify the left-hand side of the equation: $\left(\frac{(0.032)^5}{0.2 \times L_e}\right)^{1/2}$ First, calculate $0.032^5$: $0.032^5 = 0.000001048576$ So the expression becomes: $\left(\frac{0.000001048576}{0.2 \times L_e}\right)^{1/2}$ We know that this equals $0.000935$, so: $\left(\frac{0.000001048576}{0.2 \times L_e}\right)^{1/2} = 0.000935$

Step 4: Square both sides to eliminate the square root

Now, square both sides to eliminate the square root: $\frac{0.000001048576}{0.2 \times L_e} = (0.000935)^2$ Calculate the square of $0.000935$: $(0.000935)^2 = 0.000000873225$ Thus, the equation becomes: $\frac{0.000001048576}{0.2 \times L_e} = 0.000000873225$

Step 5: Solve for $L_e$

Now, solve for $L_e$: $0.000001048576 = 0.000000873225 \times 0.2 \times L_e$ $0.000001048576 = 0.000000174645 \times L_e$ Now, divide both sides by $0.000000174645$: $L_e = \frac{0.000001048576}{0.000000174645}$ $L_e \approx 6.0$

Final Answer:

$L_e \approx 6$

Would you like more details on the solution or have further questions? Here are some related questions you may find useful:

1. How would the result change if we used different values for the constants?
2. What is the significance of squaring both sides of the equation?
3. How can we check the validity of this equation with approximate values?
4. How does the choice of constants affect the outcome of such equations?
5. What happens to the solution if the values of $0.032$ and $0.06$ were significantly larger or smaller?

Tip: When working with complex equations, always simplify both sides step by step and check intermediate values for consistency.