To simplify the radical \( \sqrt[3]{16xy^6} \), we need to break down the expression under the cube root.
- First, we can factor \( 16 \) and \( y^6 \):
- \( 16 = 2^4 \)
- \( y^6 = (y^2)^3 \)
So, we rewrite the expression under the cube root: \[ \sqrt[3]{16xy^6} = \sqrt[3]{2^4 \cdot x \cdot (y^2)^3} \]
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This allows us to break it into separate radicals: \[ \sqrt[3]{2^4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{(y^2)^3} \]
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Now calculate each part:
- \( \sqrt[3]{(y^2)^3} = y^2 \)
- \( \sqrt[3]{2^4} = \sqrt[3]{16} = 2 \cdot \sqrt[3]{2} \)
So, the overall expression becomes: \[ 2y^2 \cdot \sqrt[3]{2} \cdot \sqrt[3]{x} = 2y^2 \cdot \sqrt[3]{2x} \]
Putting this together, we arrive at the simplified form: \[ 2y^2 \cdot \sqrt[3]{2x} \]
However, none of the provided answer choices matches this explicit format exactly (though it might match depending on the notation). Thus, if we use the simplest form based on the radicals given, the best option is:
\[ 2y^2 \sqrt[3]{2x} \]
Now, among the options:
- 2y²√[3]{2x} corresponds to what we derived.
- 4y²√[3]{x} includes the wrong factors.
- 2xy²√[3]{2} maintains some elements but introduces the wrong coefficient.
- 8y²√[3]{2x} does not match.
If these represent the features of the cube roots correctly, the most akin matching choice to our conclusion based on the first part is:
2y²√[3]{2x} (make sure the formatting matches what's intended).
If the response is rooted in some monkeyed answer format, please check the question's context for clarity on notation usage. Thus, the simplification approaches leaped to describe \( 2y^2 \times \sqrt[3]{2x} \).
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