The expression given is:

[ -3 \sqrt[3]{256x^4} - \sqrt[3]{32x^4} ]

Let's break it down step-by-step.

Step 1: Simplify each cube root term

We need to simplify the cube roots of $256x^4$ and $32x^4$.

1. First term: [ \sqrt[3]{256x^4} = \sqrt[3]{256} \cdot \sqrt[3]{x^4} ] We know that $256 = 2^8$, so: [ \sqrt[3]{256} = \sqrt[3]{2^8} = 2^{8/3} = 2^2 \cdot \sqrt[3]{2} = 4\sqrt[3]{2} ] And: $\sqrt[3]{x^4} = x^{4/3}$ Therefore: [ \sqrt[3]{256x^4} = 4x^{4/3}\sqrt[3]{2} ]

2. Second term: [ \sqrt[3]{32x^4} = \sqrt[3]{32} \cdot \sqrt[3]{x^4} ] We know that $32 = 2^5$, so: [ \sqrt[3]{32} = \sqrt[3]{2^5} = 2^{5/3} ] And we already have $\sqrt[3]{x^4} = x^{4/3}$, so: $\sqrt[3]{32x^4} = 2^{5/3} \cdot x^{4/3}$

Step 2: Substitute and simplify the entire expression

Now, substituting the simplified cube roots back into the original expression:

$-3(4x^{4/3} \sqrt[3]{2}) - (2^{5/3} x^{4/3})$

Distribute the constants: $-12x^{4/3} \sqrt[3]{2} - 2^{5/3}x^{4/3}$

Step 3: Factor out the common terms

We can factor out $x^{4/3}$:

$x^{4/3} \left( -12 \sqrt[3]{2} - 2^{5/3} \right)$

This is the simplified form of the given expression.

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Related questions:

1. How do you simplify higher-order roots of variables?
2. What are cube roots of common powers of 2?
3. How can you factor expressions with cube roots?
4. How would the process change if the exponents were different?
5. What happens when the variable represents a negative number?

Tip: Always try to express cube roots in terms of their simplest power of base numbers to help in factoring.