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?

---

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.

Explore how to simplify the cube root of 81x^6y^8 step-by-step. This algebraic problem involves breaking down cube roots, applying exponent rules, and simplifying radicals. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation