To simplify the given expression:

[ \sqrt[5]{\frac{x^5}{y}} \cdot 4 \cdot \sqrt[9]{\frac{2x}{y^8}} ]

we'll simplify each term step-by-step.

Step 1: Break down each term.

1. First Radical: $\sqrt[5]{\frac{x^5}{y}}$: [ \sqrt[5]{\frac{x^5}{y}} = \frac{\sqrt[5]{x^5}}{\sqrt[5]{y}} ] Since $\sqrt[5]{x^5} = x$, this simplifies to: $\frac{x}{\sqrt[5]{y}}$

2. Second Radical: $\sqrt[9]{\frac{2x}{y^8}}$: [ \sqrt[9]{\frac{2x}{y^8}} = \frac{\sqrt[9]{2x}}{\sqrt[9]{y^8}} ] For the denominator, $\sqrt[9]{y^8}$ remains as it is. So this term becomes: [ \frac{\sqrt[9]{2x}}{\sqrt[9]{y^8}} ]

Step 2: Combine both radicals.

The full expression becomes: [ 4 \cdot \frac{x}{\sqrt[5]{y}} \cdot \frac{\sqrt[9]{2x}}{\sqrt[9]{y^8}} ] Combine terms into one fraction: [ \frac{4x \cdot \sqrt[9]{2x}}{\sqrt[5]{y} \cdot \sqrt[9]{y^8}} ]

Step 3: Simplify the denominator.

Multiply the roots of $y$: [ \sqrt[5]{y} \cdot \sqrt[9]{y^8} ] Using the property of exponents: $y^{\frac{1}{5}} \cdot y^{\frac{8}{9}}$ Add the exponents: $y^{\frac{1}{5} + \frac{8}{9}}$ Convert to a common denominator: $\frac{1}{5} = \frac{9}{45}, \quad \frac{8}{9} = \frac{40}{45}$ $\frac{1}{5} + \frac{8}{9} = \frac{49}{45}$ So the denominator becomes: $y^{\frac{49}{45}}$

Step 4: Simplify the numerator.

The numerator is: $4x \cdot \sqrt[9]{2x}$ Leave as is for now.

Final Simplified Expression:

$\frac{4x \cdot \sqrt[9]{2x}}{y^{\frac{49}{45}}}$

Let me know if you'd like further clarification or alternate steps!

Related Questions:

1. How do you handle fractional exponents when simplifying radicals?
2. What are the rules for multiplying roots with different indices?
3. Can you further simplify $\sqrt[9]{2x}$ into exponential form?
4. How do you rationalize denominators involving fractional powers?
5. What are practical applications of simplifying such expressions?

Tip:

Always convert radicals into fractional exponents for easier manipulation when combining or simplifying.