To solve for \( L_1 \) in the equation 

\[ k = \frac{M}{\sqrt{L_1 \cdot L_2}} \]

we will isolate \( L_1 \). Here are the steps:

1. Start with the given equation:

   \[ k = \frac{M}{\sqrt{L_1 \cdot L_2}} \]

2. Square both sides to eliminate the square root:

   \[ k^2 = \left( \frac{M}{\sqrt{L_1 \cdot L_2}} \right)^2 \]
   \[ k^2 = \frac{M^2}{L_1 \cdot L_2} \]

3. Multiply both sides by \( L_1 \cdot L_2 \) to isolate \( M^2 \):

   \[ k^2 \cdot L_1 \cdot L_2 = M^2 \]

4. Divide both sides by \( k^2 \cdot L_2 \) to solve for \( L_1 \):

   \[ L_1 = \frac{M^2}{k^2 \cdot L_2} \]

Therefore, the solution for \( L_1 \) is:

\[ L_1 = \frac{M^2}{k^2 \cdot L_2} \]

Would you like any further details or have any questions?

Here are some related questions to expand your understanding:

1. How do you solve for \( L_2 \) in the same equation?
2. What is the physical significance of \( k \), \( M \), \( L_1 \), and \( L_2 \) in a practical context?
3. How does changing \( M \) affect \( L_1 \) if \( k \) and \( L_2 \) are constants?
4. How does changing \( k \) affect \( L_1 \) if \( M \) and \( L_2 \) are constants?
5. What happens to \( L_1 \) if \( L_2 \) is doubled, assuming \( k \) and \( M \) are constants?
6. Can \( L_1 \) be negative? Why or why not?
7. How would you graph \( L_1 \) as a function of \( k \) given fixed values of \( M \) and \( L_2 \)?
8. What are some real-world applications where this formula might be used?

**Tip:** Always check the units of measurement when working with physical equations to ensure consistency and correctness.

Learn how to solve the equation k = M / √(L1 * L2) for L1 with a detailed step-by-step solution. This problem covers key algebraic concepts and formulas related to solving equations with square roots. Suitable for advanced high school students.

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