Simplify the cube root expression: $\sqrt[3]{\frac{8x^8y^9}{27}}$.

1. Problem Description

Simplify the cube root expression: $\sqrt[3]{\frac{8x^8y^9}{27}}$.

2. Solution Steps

First, we can rewrite the cube root of a fraction as the fraction of the cube roots:

$\sqrt[3]{\frac{8x^8y^9}{27}} = \frac{\sqrt[3]{8x^8y^9}}{\sqrt[3]{27}}$

We know that $\sqrt[3]{8}=2$ and $\sqrt[3]{27}=3$. So we can write:

$\frac{\sqrt[3]{8x^8y^9}}{\sqrt[3]{27}} = \frac{2\sqrt[3]{x^8y^9}}{3}$

Now we simplify the cube root of $x^8y^9$. We know that $x^8 = x^6 \cdot x^2$ and $y^9 = (y^3)^3$. Thus we have:

$\sqrt[3]{x^8y^9} = \sqrt[3]{x^6 \cdot x^2 \cdot y^9} = \sqrt[3]{(x^2)^3 \cdot x^2 \cdot (y^3)^3} = x^2y^3\sqrt[3]{x^2}$

Plugging this back into our expression:

$\frac{2\sqrt[3]{x^8y^9}}{3} = \frac{2x^2y^3\sqrt[3]{x^2}}{3}$

3. Final Answer

$\frac{2x^2y^3\sqrt[3]{x^2}}{3}$