To simplify \( \sqrt[3]{-500x^3y^3} \), we will break it down step by step.

### Step 1: Simplify the cube root of the constants
We can split the expression inside the cube root into two parts: the numerical constant \(-500\) and the variables \(x^3y^3\).

- First, simplify the cube root of \(-500\):
\[
\sqrt[3]{-500} = -\sqrt[3]{500}
\]
- \(500\) can be factored as \(500 = 5^3 \times 2\), so:
\[
\sqrt[3]{500} = \sqrt[3]{5^3 \times 2} = 5\sqrt[3]{2}
\]
Thus, the cube root of \(-500\) is:
\[
\sqrt[3]{-500} = -5\sqrt[3]{2}
\]

### Step 2: Simplify the cube root of the variables
For the variables \(x^3\) and \(y^3\), since the powers are multiples of 3, we can directly take their cube roots:
\[
\sqrt[3]{x^3} = x \quad \text{and} \quad \sqrt[3]{y^3} = y
\]

### Step 3: Combine everything
Now, combining all the simplified parts together:
\[
\sqrt[3]{-500x^3y^3} = -5xy\sqrt[3]{2}
\]

Thus, the simplified form is:
\[
\boxed{-5xy\sqrt[3]{2}}
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you simplify other cube roots of negative numbers?
2. Can you simplify \( \sqrt[3]{54x^6y^9} \)?
3. What is the general process for simplifying cube roots of expressions with variables?
4. How do cube roots interact with powers of variables?
5. How would you solve equations involving cube roots?

**Tip:** When dealing with cube roots of negative numbers, remember that the cube root of a negative number remains negative.

Learn how to simplify the cube root of -500x³y³ with a clear, step-by-step solution. This problem focuses on algebraic simplification of radicals, properties of cube roots, and exponent rules. Suitable for high school students in Grades 9-11.

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