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).