To simplify the expression, we need to combine like terms for the radical terms. Here’s the expression provided:

[ 10 \sqrt[3]{3x} - 2 \sqrt{3x} + 6 \sqrt{12x} - 3 \sqrt[3]{24x} ]

Let's simplify each term and see if any terms can be combined.

1. Term Analysis:

   • $10 \sqrt[3]{3x}$: This term is in terms of the cube root of $3x$.
   • $-2 \sqrt{3x}$: This term is in terms of the square root of $3x$.
   • $6 \sqrt{12x}$: We can simplify $\sqrt{12x}$ by breaking down $12x$ into factors.
   • $-3 \sqrt[3]{24x}$: We can simplify $\sqrt[3]{24x}$ by breaking down $24x$ into factors.

2. Simplify Each Term:

   • $10 \sqrt[3]{3x}$: This remains as is.
   • $-2 \sqrt{3x}$: This remains as is.
   • $6 \sqrt{12x} = 6 \cdot \sqrt{4 \cdot 3x} = 6 \cdot 2 \sqrt{3x} = 12 \sqrt{3x}$.
   • $-3 \sqrt[3]{24x} = -3 \cdot \sqrt[3]{8 \cdot 3x} = -3 \cdot 2 \sqrt[3]{3x} = -6 \sqrt[3]{3x}$.

3. Combine Like Terms:

   • For the terms involving $\sqrt[3]{3x}$: $10 \sqrt[3]{3x} - 6 \sqrt[3]{3x} = 4 \sqrt[3]{3x}$.
   • For the terms involving $\sqrt{3x}$: $-2 \sqrt{3x} + 12 \sqrt{3x} = 10 \sqrt{3x}$.

4. Final Simplified Expression: $4 \sqrt[3]{3x} + 10 \sqrt{3x}$

This is the simplest form by combining like radical terms.

Would you like more details on any part of this solution?

Here are five related questions to expand your understanding:

1. How would the simplification change if all terms were in terms of square roots instead of mixed roots?
2. Can you simplify the expression further if different values for $x$ are assumed?
3. How do you combine terms when the radicals have different coefficients?
4. How would the process change if one of the terms had an additional constant multiplier?
5. What happens if you have cube roots and square roots of different bases?

Tip: When simplifying radicals, always look to factor out squares or cubes to combine terms effectively.