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The given expression is the cube root of \( 24x^8y^6 \). We will simplify this step by step:

### Expression:
\[
\sqrt[3]{24x^8y^6}
\]

### Step 1: Factor the constant inside the cube root
First, we simplify the constant \( 24 \) by factoring it:
\[
24 = 2^3 \times 3
\]
So, the expression becomes:
\[
\sqrt[3]{2^3 \times 3 \times x^8 \times y^6}
\]

### Step 2: Apply cube root to each factor
We can now apply the cube root to each part:
\[
\sqrt[3]{2^3} \times \sqrt[3]{3} \times \sqrt[3]{x^8} \times \sqrt[3]{y^6}
\]

1. \( \sqrt[3]{2^3} = 2 \)
2. \( \sqrt[3]{3} \) remains as \( \sqrt[3]{3} \) since 3 is not a perfect cube.
3. \( \sqrt[3]{x^8} = x^{8/3} = x^2 \times x^{2/3} \)
4. \( \sqrt[3]{y^6} = y^{6/3} = y^2 \)

### Step 3: Combine the simplified terms
The final simplified expression is:
\[
2x^2y^2 \times \sqrt[3]{3x^2}
\]

So, the simplified form of \( \sqrt[3]{24x^8y^6} \) is:
\[
2x^2y^2 \cdot \sqrt[3]{3x^2}
\]

Would you like more details on any step?

Here are 5 related questions to expand your understanding:
1. How do you simplify cube roots for other expressions?
2. How can you handle cube roots for variables with fractional exponents?
3. What is the process for simplifying higher-order roots (e.g., fourth roots)?
4. How do you handle cube roots of negative numbers?
5. What are the differences between square roots and cube roots in simplification?

**Tip**: When simplifying roots of variables with exponents, divide the exponent by the root (e.g., for a cube root, divide the exponent by 3).

Simplify the expression \(\sqrt[3]{24x^8y^6}\).

This problem involves simplifying the cube root of \(24x^8y^6\), using exponent laws and radical properties. The solution shows step-by-step factoring and simplification of both constants and variables. Suitable for students in Grades 9-12.

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