Let's simplify the expression $4\sqrt[3]{128x^3} + \sqrt[3]{16x^3}$.

Step 1: Simplify each cube root separately

1. $\sqrt[3]{128x^3}$

   • Factor 128: $128 = 2^7$
   • So, $\sqrt[3]{128x^3} = \sqrt[3]{2^7 \cdot x^3} = 2^2 \cdot x \cdot \sqrt[3]{2} = 4x\sqrt[3]{2}$

2. $\sqrt[3]{16x^3}$

   • Factor 16: $16 = 2^4$
   • So, $\sqrt[3]{16x^3} = \sqrt[3]{2^4 \cdot x^3} = 2 \cdot x \cdot \sqrt[3]{2} = 2x\sqrt[3]{2}$

Step 2: Combine the simplified terms

Now, substitute these into the original expression:

[ 4\sqrt[3]{128x^3} + \sqrt[3]{16x^3} = 4 \times (4x\sqrt[3]{2}) + 2x\sqrt[3]{2} ]

This simplifies to:

[ 16x\sqrt[3]{2} + 2x\sqrt[3]{2} ]

Step 3: Factor out the common terms

Factor out $2x\sqrt[3]{2}$ from the expression:

[ 2x\sqrt[3]{2}(8 + 1) = 18x\sqrt[3]{2} ]

So, the simplified expression is:

$\boxed{18x\sqrt[3]{2}}$

Would you like more details on this simplification? Here are some related questions:

1. How do you simplify cube roots in general?
2. What is the process for factoring out common terms?
3. How would the simplification change if the expression were different?
4. Can this method be applied to higher roots, like the 4th root?
5. What are some real-world applications of cube roots in mathematics?

Tip: Always check if the radicand (the number inside the root) can be factored into smaller parts that include perfect powers for easier simplification.