The given expression is:

[ \sqrt[3]{y^x} \cdot \sqrt[3]{y^{2x}} ]

To simplify this, we can use the property of radicals and exponents. Specifically, for cube roots:

[ \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} ]

Applying this property:

[ \sqrt[3]{y^x} \cdot \sqrt[3]{y^{2x}} = \sqrt[3]{y^x \cdot y^{2x}} ]

Since $y^x \cdot y^{2x} = y^{x + 2x} = y^{3x}$, we now have:

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

The cube root of $y^{3x}$ is $y^x$. Therefore, the simplified expression is:

$y^x$

Would you like any further details or clarifications?

Here are some related questions to explore:

1. How can we generalize the rule for multiplying radicals with different bases?
2. What happens if the index of the root is different (e.g., 4th or 5th root)?
3. How would the solution change if $x$ were negative or a fraction?
4. Can you simplify expressions with different radical indices without converting them?
5. How does this radical simplification connect to logarithms?

Tip: When simplifying radicals, always look for opportunities to combine exponents before applying root properties!