https://file.aiquickdraw.com/mathbot/file/1727720755438z3o3w5xm.jpg

The expression provided in the image is:

\[
\sqrt[3]{81a^3}
\]

We are asked to simplify this expression, assuming all variables represent positive numbers.

### Steps to Simplify:

1. Break down the constant \(81\) into its prime factors:
   \[
   81 = 3^4
   \]

2. Rewrite the entire expression using this factorization:
   \[
   \sqrt[3]{(3^4) \cdot (a^3)}
   \]

3. Apply the cube root to both terms:
   - For \(3^4\), recall that \(\sqrt[3]{3^4} = 3^{4/3}\), which simplifies to:
     \[
     3^{4/3} = 3 \cdot \sqrt[3]{3}
     \]
   - For \(a^3\), we know that \(\sqrt[3]{a^3} = a\).

4. Combine the results:
   \[
   3a \cdot \sqrt[3]{3}
   \]

So, the simplified form of the expression is:
\[
3a \cdot \sqrt[3]{3}
\]

Would you like more details or have any questions?

Here are five related questions to consider:
1. How would this process change for a square root instead of a cube root?
2. What happens if \( a \) is a negative number? Does the simplification still hold?
3. Can you simplify expressions with higher powers, like the 5th or 7th root, similarly?
4. How would you simplify an expression with multiple variables, such as \(\sqrt[3]{27b^6c^9}\)?
5. How can you apply this method to solve radical equations?

**Tip**: Breaking numbers into their prime factors often helps when simplifying radicals!

Simplify the radical: cube root of (81a^3).

Learn how to simplify the cube root of 81a^3 by breaking down constants into prime factors and applying the properties of exponents and radicals. Step-by-step solution suitable for high school students (Grades 9-11).

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