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The equation given is:

\[
\frac{81}{x} = \frac{x}{y} = \frac{y}{z} = \frac{z}{256} = k
\]

We need to calculate \(x + y + z\).

### Step-by-Step Solution:

1. **Express each variable in terms of \(k\):**
   - \( \frac{81}{x} = k \)  ⇒ \( x = \frac{81}{k} \)
   - \( \frac{x}{y} = k \)  ⇒ \( y = \frac{x}{k} \)
   - \( \frac{y}{z} = k \)  ⇒ \( z = \frac{y}{k} \)
   - \( \frac{z}{256} = k \) ⇒ \( z = 256k \)

2. **Substitute the expressions:**
   - Start with \( z = 256k \).
   - Then, using \( z = \frac{y}{k} \), we get \( y = 256k^2 \).
   - Finally, using \( y = \frac{x}{k} \), we get \( x = 256k^3 \).

3. **Use \( x = \frac{81}{k} \):**
   - We have two expressions for \( x \): \( x = \frac{81}{k} \) and \( x = 256k^3 \).
   - Set them equal to each other: \( \frac{81}{k} = 256k^3 \).
   - Multiply both sides by \( k \) to clear the fraction: \( 81 = 256k^4 \).
   - Solve for \( k^4 \): \( k^4 = \frac{81}{256} \).
   - Take the fourth root: \( k = \frac{3}{4} \).

4. **Find \( x \), \( y \), and \( z \) values:**
   - \( x = \frac{81}{k} = \frac{81}{\frac{3}{4}} = 108 \)
   - \( y = 256k^2 = 256 \times \left(\frac{3}{4}\right)^2 = 144 \)
   - \( z = 256k = 256 \times \frac{3}{4} = 192 \)

5. **Sum the values:**
   \[
   x + y + z = 108 + 144 + 192 = 444
   \]

### Final Answer:
The sum \( x + y + z \) is \( \boxed{444} \).

Would you like more details on any step? 

Here are five related questions:
1. How would you solve the problem if the constant \( 256 \) were different?
2. What would change in the process if you were given an additional equation?
3. How would you check if the calculated values of \(x\), \(y\), and \(z\) satisfy all parts of the original equation?
4. Could this problem be solved graphically, and if so, how?
5. How would the solution change if \( k \) was directly given as a value?

**Tip:** Always double-check your substitutions and calculations, especially when dealing with multiple expressions for variables.

Learn how to solve a proportional equation problem where 81/x = x/y = y/z = z/256. This problem involves calculating values of x, y, and z based on proportional relationships. Suitable for advanced high school students.

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