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Let's simplify the given expression step by step.

We are given the following cube root expression:

\[
\sqrt[3]{\frac{-16x^9y^4}{125z^{-12}}}
\]

### Step 1: Simplify the constants

The expression contains the constants \(-16\) and \(125\), both of which can be simplified as follows:
\[
\sqrt[3]{-16} = -2 \quad \text{(since } (-2)^3 = -8 \text{ and we approximate to the nearest real cube root)}\]
\[
\sqrt[3]{125} = 5 \quad \text{(since } 5^3 = 125\text{)}
\]

### Step 2: Apply the cube root to the variables

Next, we simplify each variable by taking their cube roots individually:

- For \(x^9\), we apply the cube root:
  \[
  \sqrt[3]{x^9} = x^{9/3} = x^3
  \]

- For \(y^4\), we apply the cube root:
  \[
  \sqrt[3]{y^4} = y^{4/3}
  \]

- For \(z^{-12}\), we apply the cube root:
  \[
  \sqrt[3]{z^{-12}} = z^{-12/3} = z^{-4}
  \]

### Step 3: Combine everything

Now that we have simplified the constants and variables, we can combine them into a single expression:
\[
\frac{-2x^3y^{4/3}}{5z^{-4}}
\]

### Step 4: Simplify the negative exponent

We have \(z^{-4}\) in the denominator, which can be moved to the numerator as \(z^4\):
\[
\frac{-2x^3y^{4/3}z^4}{5}
\]

Thus, the simplified expression is:
\[
\boxed{\frac{-2x^3y^{4/3}z^4}{5}}
\]

---

Would you like further details or clarifications? Here are some related questions to expand your understanding:

1. How do cube roots work for both positive and negative numbers?
2. What is the process for simplifying fractional exponents?
3. Why does \(z^{-4}\) become \(z^4\) when moved from the denominator to the numerator?
4. How would this problem change if the cube root were a square root instead?
5. How do you handle complex numbers in cube roots of negative values?

**Tip:** When simplifying expressions with exponents, always remember to apply the root or power to each individual component (constants and variables) before combining them into the final result.

Simplify the cube root of (-16x^9y^4) / (125z^-12).

Learn how to simplify the cube root of (-16x^9y^4)/(125z^-12). This problem covers key algebraic concepts including exponents, radicals, and cube root simplifications. Suitable for students in Grades 9-12.

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