To solve the given expression $\sqrt[3]{81y^8}$, let’s break it down step by step:

Step 1: Factorize the base

The cube root is applied to both $81$ and $y^8$: [ \sqrt[3]{81y^8} = \sqrt[3]{81} \cdot \sqrt[3]{y^8}. ]

Step 2: Simplify the cube root of 81

$81 = 3^4$, so: [ \sqrt[3]{81} = \sqrt[3]{3^4} = 3^{4/3} = 3 \cdot \sqrt[3]{3}. ]

Step 3: Simplify the cube root of $y^8$

Using the property $\sqrt[3]{y^8} = y^{8/3}$, separate the integer and fractional parts: $y^{8/3} = y^2 \cdot \sqrt[3]{y^2}.$

Step 4: Combine the results

Now, multiply everything: [ \sqrt[3]{81y^8} = 3 \cdot y^2 \cdot \sqrt[3]{3y^2}. ]

Final Answer:

$3y^2\sqrt[3]{3y^2}.$

Let me know if you want further clarification or detailed steps.

Related Questions:

1. What are the rules for simplifying radicals with variables?
2. How do fractional exponents relate to radical expressions?
3. Can the cube root of any number always be expressed in simpler terms?
4. How do cube roots differ from square roots in terms of properties?
5. How can simplifying radicals be used in real-world applications?

Tip:

Always express powers and roots clearly and ensure variables are positive to avoid unnecessary steps!