The expression $\sqrt[3]{81x^6y^8}$ can be simplified as follows:

1. First, break down the cube root into individual components: [ \sqrt[3]{81x^6y^8} = \sqrt[3]{81} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^8} ]

2. Simplify each component:

   • $\sqrt[3]{81}$: Since 81 = $3^4$, we can simplify $\sqrt[3]{81}$ as: [ \sqrt[3]{81} = \sqrt[3]{3^4} = 3^{4/3} ]
   • $\sqrt[3]{x^6}$: Apply the cube root to the exponent: $\sqrt[3]{x^6} = x^{6/3} = x^2$
   • $\sqrt[3]{y^8}$: Apply the cube root to the exponent: $\sqrt[3]{y^8} = y^{8/3}$

3. Now combine the simplified components: $3^{4/3} \cdot x^2 \cdot y^{8/3}$

Thus, the simplified form of $\sqrt[3]{81x^6y^8}$ is: $3^{4/3} x^2 y^{8/3}$

Would you like more details or have any questions on this?

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Here are 5 related questions:

1. How do you simplify the cube root of a fraction?
2. What is the cube root of a higher degree polynomial?
3. How do you calculate $\sqrt[3]{x^9}$ in general form?
4. How does the cube root affect negative exponents?
5. What are the rules for fractional exponents?

Tip: When simplifying cube roots, express the terms as powers to apply exponent rules efficiently.