https://file.aiquickdraw.com/mathbot/file/17256776846895qd9jisb.jpg

The equation in the image is a complex mathematical expression. I will break it down and solve or verify the given equation.

The equation appears to be:

\[
\frac{\left( x^2 - \frac{1}{y^2} \right)^x \left( \frac{-1}{y} \right)^{y-x}}{\left( y^2 - \frac{1}{x^2} \right)^y \left( y + \frac{1}{x} \right)^{x-y}} = \left( \frac{x}{y} \right)^{x+y}
\]

Let me analyze and simplify each part of the expression.

### Left Side:
1. The numerator: \(\left( x^2 - \frac{1}{y^2} \right)^x \cdot \left( \frac{-1}{y} \right)^{y-x}\)
2. The denominator: \(\left( y^2 - \frac{1}{x^2} \right)^y \cdot \left( y + \frac{1}{x} \right)^{x-y}\)

### Right Side:
The right side is \(\left( \frac{x}{y} \right)^{x+y}\).

I will now check if both sides are equal through simplification.The equation simplifies to the following form:

\[
\left( \frac{x}{y} \right)^{x+y} = (-1/y)^{y-x} \cdot \left( \frac{x^2 y^2 - 1}{x^2} \right)^{-y} \cdot \left( \frac{xy + 1}{x} \right)^{y-x} \cdot \left( \frac{x^2 y^2 - 1}{y^2} \right)^x
\]

This shows that the left-hand side and the right-hand side are not directly equal without further simplification, though there is a relationship between them. The form involves powers and multiplicative factors that might require more specific values of \(x\) and \(y\) to fully verify equality.

Would you like me to check for specific values of \(x\) and \(y\), or explore further details on simplifying these terms?

### Relative questions:
1. How would different values of \(x\) and \(y\) affect the equation?
2. Can you identify patterns in exponents between both sides of the equation?
3. Are there any assumptions or constraints that simplify this expression further?
4. How do fractional powers in these expressions impact the solution?
5. Can trigonometric or logarithmic substitutions help in solving this?

### Tip:
When dealing with complex exponents, checking for particular values or conditions can sometimes reveal simplifications that are not obvious algebraically.

Simplify the given complex mathematical expression and verify the equality.

Learn how to simplify and verify complex exponential expressions involving multiple terms and exponents. This example demonstrates the application of exponent rules and simplification techniques, suitable for advanced high school students.

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