We are tasked with simplifying the following expression:

[ -3 \sqrt[3]{625x^8} + \sqrt[3]{135x^8} ]

Step-by-Step Simplification:

1. Simplify each cube root term:

   • Start with $\sqrt[3]{625x^8}$:

     • $625 = 5^4$, so rewrite the cube root: [ \sqrt[3]{625x^8} = \sqrt[3]{5^4 \cdot x^8} = 5 \cdot x^{8/3} ] This simplifies to $5x^{8/3}$.

   • Now simplify $\sqrt[3]{135x^8}$:

     • Factor $135 = 5 \cdot 27$, and since $27 = 3^3$, rewrite: [ \sqrt[3]{135x^8} = \sqrt[3]{5 \cdot 3^3 \cdot x^8} = 3 \cdot \sqrt[3]{5x^8} ] Next, simplify $\sqrt[3]{5x^8}$, which becomes $5^{1/3} x^{8/3}$.

2. Substitute back into the original expression: $-3(5x^{8/3}) + 3(5^{1/3}x^{8/3})$ Factor out $x^{8/3}$ and 3: $3x^{8/3}(-5 + 5^{1/3})$

Thus, the simplified expression is:

$3x^{8/3}(5^{1/3} - 5)$

Let me know if you'd like further details on the steps!

Related questions:

1. How do you simplify higher roots of expressions with exponents?
2. What is the general method for simplifying cube roots involving variables?
3. How do you factor cube roots with numerical coefficients like 625 or 135?
4. What happens if the exponents of $x$ were different from 8 in each term?
5. How do cube roots behave differently for negative numbers?

Tip: When simplifying expressions involving cube roots, always start by factoring the numbers inside the root into prime factors. This often makes it easier to identify powers that can be simplified directly.