To simplify the expression \( \sqrt[3]{16xy^6} \), we can break it down as follows:
- Factor \( 16 \), which is \( 2^4 \).
- The variable \( y^6 \) can be rewritten as \( (y^2)^3 \).
Now, we have:
\[ \sqrt[3]{16xy^6} = \sqrt[3]{2^4 \cdot x \cdot (y^2)^3} \]
Using the properties of radicals, we can separate this into different parts:
\[ = \sqrt[3]{2^4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{(y^2)^3} \]
Remember that \( \sqrt[3]{(y^2)^3} = y^2 \):
\[ = y^2 \cdot \sqrt[3]{2^4} \cdot \sqrt[3]{x} \]
Now we simplify \( \sqrt[3]{2^4} \):
\[ \sqrt[3]{2^4} = 2^{4/3} = 2^{1 + 1/3} = 2 \sqrt[3]{2} \]
Putting it all together, we have:
\[ = y^2 \cdot \left( 2 \sqrt[3]{2} \right) \cdot \sqrt[3]{x} \]
Finally, this can be expressed as:
\[ = 2y^2 \sqrt[3]{2x} \]
Therefore, the correct response is:
2y²∛(2x)
None of the provided options exactly match this, but assuming there's a formatting issue, the closest presentation would be 2y²∛2x. If there was another option using radical notation for cubic roots, that could help as well. Always double-check for any notation discrepancies.
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