This equation appears to be a complex fraction involving nested terms and various mathematical operations. I will simplify and solve it step by step.

The equation can be summarized as:

$\frac{\left(1 + \frac{x}{y}\right)^{\frac{x}{x-y}} \times \left(1 - \frac{x}{y}\right)^{\frac{y}{x-y}}}{\left(\frac{y}{x} + 1\right)^{\frac{x}{x-y}} \times \left(\frac{y}{x} - 1\right)^{\frac{y}{x-y}}}.$

Steps to simplify:

1. Rewrite and identify key terms:

   • Numerator: $\left(1 + \frac{x}{y}\right)^{\frac{x}{x-y}} \times \left(1 - \frac{x}{y}\right)^{\frac{y}{x-y}}.$
   • Denominator: $\left(\frac{y}{x} + 1\right)^{\frac{x}{x-y}} \times \left(\frac{y}{x} - 1\right)^{\frac{y}{x-y}}.$

2. Simplify the terms:

   • For the terms involving $1 + \frac{x}{y}$ and $1 - \frac{x}{y}$, note that $1 + \frac{x}{y} = \frac{y+x}{y}$ and $1 - \frac{x}{y} = \frac{y-x}{y}$.
   • For the denominator terms, rewrite $\frac{y}{x} + 1 = \frac{y+x}{x}$ and $\frac{y}{x} - 1 = \frac{y-x}{x}$.

3. Combine and factorize: Substitute the simplified forms back into the equation: $\frac{\left(\frac{y+x}{y}\right)^{\frac{x}{x-y}} \times \left(\frac{y-x}{y}\right)^{\frac{y}{x-y}}}{\left(\frac{y+x}{x}\right)^{\frac{x}{x-y}} \times \left(\frac{y-x}{x}\right)^{\frac{y}{x-y}}}.$

4. Use properties of exponents: Separate the powers for $y+x$ and $y-x$:

   • Numerator: $\frac{(y+x)^{\frac{x}{x-y}} \cdot (y-x)^{\frac{y}{x-y}}}{y^{\frac{x}{x-y}} \cdot y^{\frac{y}{x-y}}}.$
   • Denominator: $\frac{(y+x)^{\frac{x}{x-y}} \cdot (y-x)^{\frac{y}{x-y}}}{x^{\frac{x}{x-y}} \cdot x^{\frac{y}{x-y}}}.$

5. Cancel common terms:

   • The terms $(y+x)^{\frac{x}{x-y}}$ and $(y-x)^{\frac{y}{x-y}}$ in the numerator and denominator cancel out.
   • You're left with: $\frac{x^{\frac{x}{x-y}} \cdot x^{\frac{y}{x-y}}}{y^{\frac{x}{x-y}} \cdot y^{\frac{y}{x-y}}}.$

6. Combine exponents: Group the terms for $x$ and $y$: $\left(\frac{x}{y}\right)^{\frac{x+y}{x-y}}.$

Final Answer:

$\boxed{\left(\frac{x}{y}\right)^{\frac{x+y}{x-y}}.}$

Let me know if you need further clarification or additional steps explained!

Here are some related questions:

1. How can properties of exponents simplify nested fractions?
2. What is the geometric interpretation of the given expression?
3. Can this equation be graphed for specific values of $x$ and $y$?
4. How does this expression behave as $x \to y$?
5. Are there conditions where this simplification is undefined (e.g., division by zero)?

Tip: When solving such problems, always check if terms in the numerator and denominator cancel to simplify the calculations.