https://file.aiquickdraw.com/mathbot/file/17267944982823hel7jnu.jpg

The problem in the image asks for the value of:

\[
(y - x)^{2026} + (x - y)^{2025}
\]

with the condition:

\[
\left( \frac{1}{4} \left( \frac{y - 1}{3} \right) \right) = \left( \frac{1}{4} \frac{x}{z} \right)
\]

### Step 1: Simplify the first part of the expression
We know that:

\[
x - y = -(y - x)
\]

This means that:

\[
(x - y)^{2025} = (-(y - x))^{2025} = - (y - x)^{2025}
\]

Now, substitute this into the original expression:

\[
(y - x)^{2026} + (x - y)^{2025} = (y - x)^{2026} - (y - x)^{2025}
\]

### Step 2: Solve the equation for \( x \), \( y \), and \( z \)
We are given the equation:

\[
\frac{1}{4} \cdot \frac{y - 1}{3} = \frac{1}{4} \cdot \frac{x}{z}
\]

Simplifying both sides, cancel out the \( \frac{1}{4} \):

\[
\frac{y - 1}{3} = \frac{x}{z}
\]

Multiply both sides by 3:

\[
y - 1 = \frac{3x}{z}
\]

From this equation, we can express \( y \) in terms of \( x \) and \( z \):

\[
y = \frac{3x}{z} + 1
\]

### Step 3: Conclusion
With the simplified expression for \( y \) and the expression:

\[
(y - x)^{2026} - (y - x)^{2025}
\]

you can plug in the values of \( y \) and \( x \) to evaluate the final result, but without specific values for \( x \) and \( z \), we cannot compute the exact numerical value. However, the key expression is simplified.

Would you like to explore specific values or need further clarification on any step? 

---

Here are 5 follow-up questions for further exploration:

1. How can the values of \( x \), \( y \), and \( z \) be determined given more conditions?
2. What is the behavior of the expression as \( y \) approaches \( x \)?
3. What happens if the powers in the expression are lower, e.g., \( y - x \) to the power of 2 or 3?
4. How does the equation change if \( z = 1 \)?
5. How can we derive general formulas for similar exponential expressions?

**Tip:** When solving equations with large exponents, look for symmetries or simplifications to reduce computation complexity.

Nilai dari (y - x)^2026 + (x - y)^2025 agar (1/4)((y - 1)/3) = (1/4)(x/z) adalah ...

Find the value of (y - x)^2026 + (x - y)^2025 under the condition (1/4)((y - 1)/3) = (1/4)(x/z). This problem demonstrates algebraic manipulation, exponent rules, and equation simplification for high school students. Suitable for Grades 10-12.

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