To simplify the expression \( \sqrt[3]{16xy^6} \), we can break it down into its factors:
- Write \( 16 \) as \( 2^4 \).
- We know that \( y^6 \) can be rewritten as \( (y^2)^3 \).
Thus, we can rewrite \( \sqrt[3]{16xy^6} \) in terms of its prime factors:
\[ \sqrt[3]{16xy^6} = \sqrt[3]{2^4 \cdot x \cdot (y^2)^3} \]
Now, we can separate the cube root:
\[ \sqrt[3]{16xy^6} = \sqrt[3]{2^4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{(y^2)^3} \]
Calculating each part:
- \( \sqrt[3]{2^4} = \sqrt[3]{16} = 2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}} = 2 \cdot \sqrt[3]{2} = 2\sqrt[3]{2} \)
- \( \sqrt[3]{x} = \sqrt[3]{x} \)
- \( \sqrt[3]{(y^2)^3} = y^2 \)
Putting it all together, we have:
\[ \sqrt[3]{16xy^6} = 2y^2\sqrt[3]{2x} \]
So, the simplified expression is:
\[ 2y^2 \sqrt[3]{2x} \]
Among the provided options, this can be matched with:
2y² cube root of 2x
The correct response would be:
2y²√3 (Following the notation could have a misunderstanding. Clarification should be based on \( \sqrt[3]{2x} \) specifically)
The correct format should probably look like:
2y²√(3) if matched correctly with the radical.
This was based on assessing the options but might be censored based on context.
1 answer