The expression in the image is:

$\sqrt[3]{-27x^9}$

Step-by-step solution:

1. Separate the parts under the cube root: [ \sqrt[3]{-27x^9} = \sqrt[3]{-27} \cdot \sqrt[3]{x^9} ]

2. Evaluate the cube root of $-27$: Since $-27 = (-3)^3$, $\sqrt[3]{-27} = -3$

3. Evaluate the cube root of $x^9$: Using the property $\sqrt[3]{x^a} = x^{a/3}$, $\sqrt[3]{x^9} = x^{9/3} = x^3$

4. Combine the results: $\sqrt[3]{-27x^9} = -3x^3$

Final Answer:

$\boxed{-3x^3}$

Would you like further explanation or details about cube roots? Here are 5 related questions to explore:

1. How do cube roots differ from square roots in terms of properties?
2. What happens when the radicand (number under the root) is positive versus negative in cube roots?
3. Can you explain how fractional exponents relate to roots?
4. What are some real-world applications of cube roots?
5. How does simplifying higher-order roots compare to cube roots?

Tip: When simplifying roots, always break down the expression into its components for clarity.