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To solve the given problem, let's analyze the expression:

\[
\sqrt{578 \cdot z^9} - \sqrt{128 \cdot z^2}
\]

We'll simplify each square root term separately:

### Step 1: Simplify \(\sqrt{578 \cdot z^9}\)
- \(\sqrt{578 \cdot z^9} = \sqrt{578} \cdot \sqrt{z^9}\).
- \(\sqrt{z^9} = z^{9/2} = z^4 \cdot \sqrt{z}\).
- Thus, \(\sqrt{578 \cdot z^9} = \sqrt{578} \cdot z^4 \cdot \sqrt{z}\).

### Step 2: Simplify \(\sqrt{128 \cdot z^2}\)
- \(\sqrt{128 \cdot z^2} = \sqrt{128} \cdot \sqrt{z^2}\).
- \(\sqrt{z^2} = z\).
- \(\sqrt{128} = \sqrt{64 \cdot 2} = 8\sqrt{2}\).
- Thus, \(\sqrt{128 \cdot z^2} = 8 \sqrt{2} \cdot z\).

### Step 3: Simplify the entire expression
Now we have:
\[
\sqrt{578 \cdot z^9} - \sqrt{128 \cdot z^2} = \sqrt{578} \cdot z^4 \cdot \sqrt{z} - 8 \sqrt{2} \cdot z
\]

To match this with the given answer choices, we should evaluate \(\sqrt{578}\). Calculating this approximately:
\[
\sqrt{578} \approx 24
\]

So the first term becomes approximately:
\[
24 \cdot z^4 \cdot \sqrt{z}
\]

However, none of the answer options directly reflect this exact form, suggesting that the question may require further approximation or factoring. Given the choices, \(15 \cdot z^4 \cdot \sqrt{2z}\) might be the best match based on possible approximations.

### Answer:
**D. \(15z^4\sqrt{2z}\)**

Would you like more details on the steps or explanations for the process? Here are some related questions:

1. How do you simplify expressions involving square roots and exponents?
2. What is the method for estimating square roots?
3. How do you factor terms when working with radicals?
4. How can you check if your solution matches an answer choice?
5. Why is it important to approximate in multiple-choice questions?

**Tip:** Always break down radicals involving variables separately from constants, as this helps in simplifying complex expressions.

Which expression is equivalent to √(578·z^9) − √(128·z^2)?

Understand how to simplify the radical expression √(578·z^9) − √(128·z^2) with step-by-step solutions. Learn key algebra concepts, such as properties of radicals and exponent rules, to identify the correct answer. Suitable for high school students in Grades 9-12.

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