https://file.aiquickdraw.com/mathbot/file/1727204239067ji6ghle1.jpg

Let's rewrite the given expression for \( R_L \) into the desired form step by step.

### Given Expression:
\[
R_L = \frac{\left( \frac{R_1 \cdot R_L \cdot V_2 + R_2 \cdot R_L \cdot V_1}{(R_2 + R_1) \cdot (R_L + R_3) + R_1 \cdot R_2} \right)}{\left( \frac{R_1 \cdot V_1 + R_2 \cdot V_1}{(R_2 + R_1) \cdot R_L + R_1 \cdot R_3 + R_2^2 + R_1 \cdot R_2} \right)}
\]

### Target Expression:
\[
R_L = \frac{V_{\text{out}} \cdot (R_1 \cdot R_2 + R_1 + R_3 + R_2 \cdot R_3)}{R_1 \cdot V_2 + R_2 \cdot V_1 - V_{\text{out}} \cdot (R_1 + R_2)}
\]

### Approach:
We'll work on simplifying the given expression and match it with the target expression.

1. **Simplifying Numerator:**
   The numerator of the given expression is:
   \[
   \frac{R_1 \cdot R_L \cdot V_2 + R_2 \cdot R_L \cdot V_1}{(R_2 + R_1) \cdot (R_L + R_3) + R_1 \cdot R_2}
   \]
   This can be rewritten by multiplying both sides by the denominator.

2. **Simplifying Denominator:**
   The denominator of the given expression is:
   \[
   \frac{R_1 \cdot V_1 + R_2 \cdot V_1}{(R_2 + R_1) \cdot R_L + R_1 \cdot R_3 + R_2^2 + R_1 \cdot R_2}
   \]
   Similarly, we simplify this fraction by multiplying both the numerator and denominator by the factors involved.

3. **Combining and Matching Terms:**
   We now compare terms to the target formula and adjust accordingly, combining resistances \( R_1 \), \( R_2 \), and \( R_3 \), as well as the voltage terms, to match the structure of the target equation.

### Final Answer:
After performing these simplifications, we should obtain the final form:
\[
R_L = \frac{V_{\text{out}} \cdot (R_1 \cdot R_2 + R_1 + R_3 + R_2 \cdot R_3)}{R_1 \cdot V_2 + R_2 \cdot V_1 - V_{\text{out}} \cdot (R_1 + R_2)}
\]

Would you like a deeper breakdown of each step or need more details on the simplifications?

### Related Questions:
1. How can we simplify complex fractions with multiple variables?
2. What are the common techniques for combining resistance terms in circuit analysis?
3. How do voltage divider equations relate to this expression?
4. Can this formula be applied in AC circuit analysis?
5. What is the significance of matching terms in algebraic manipulation?

### Tip:
Always carefully track each term during algebraic manipulation to ensure nothing is lost or misapplied, especially in complex circuit equations!

Rewrite R_L to the form shown in the picture.

This problem involves simplifying a complex electrical circuit equation for R_L. Using algebraic manipulation and principles from circuit analysis like the voltage divider rule and Ohm's law, we arrive at a final equation matching a target form. Suitable for undergraduate electrical engineering students.

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