To simplify the expression \(\sqrt[3]{16xy^6}\), we can begin by breaking down the components inside the cube root.
- Factor the components:
- \(16\) can be expressed as \(2^4\).
- \(x\) stays as \(x\).
- \(y^6\) is already a perfect cube since \(y^6 = (y^2)^3\).
So we rewrite the expression inside the cube root: \[ 16xy^6 = 2^4 x y^6 = 2^3 \cdot 2 \cdot x \cdot (y^2)^3 \]
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Apply the cube root: Now we can take the cube root: \[ \sqrt[3]{16xy^6} = \sqrt[3]{2^3 \cdot 2 \cdot x \cdot (y^2)^3} \] This can be simplified as follows: \[ = \sqrt[3]{2^3} \cdot \sqrt[3]{(y^2)^3} \cdot \sqrt[3]{2} \cdot \sqrt[3]{x} \] \[ = 2 \cdot y^2 \cdot \sqrt[3]{2} \cdot \sqrt[3]{x} \]
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Combine under a single radical: We can express that as: \[ = 2y^2 \sqrt[3]{2x} \]
Thus, the simplified form of \(\sqrt[3]{16xy^6}\) is: \[ \boxed{2y^2 \sqrt[3]{2x}} \]
So the original expression \(16xy^6^{\frac{1}{3}} = 2y^2 \sqrt[3]{2x}\).