To rewrite \(-16\sqrt[3]{3}\) as a product of an integer and an irrational cube root, we first notice that \(-16\) can be expressed as \(-4 \cdot 4\).
We take \(-4\) out since it's an integer, and we can keep the cube root of \(3\) intact. This gives us:
\[ -16\sqrt[3]{3} = -4 \cdot 4\sqrt[3]{3} \]
Next, we simplify the expression. Since \(4\) can be factored out, we can rewrite \(4 = 2^2\), so:
\[ 4\sqrt[3]{3} = 4\sqrt[3]{3} = 2^2 \sqrt[3]{3} \]
This does not change the form significantly in this case. In order to bring the expression into the format requested, we set it as:
\[ -4 \times \sqrt[3]{4 \cdot 3} = -4 \times \sqrt[3]{12} \]
However, this problem specifically asks for it as a product with a cube root, and referring back, we note:
The integer is \(-4\), and we have \( \sqrt[3]{4}\sqrt[3]{3}\) = \(\sqrt[3]{12}\).
Thus, we can keep the integer part as \(-4\) but we have:
Final answer would remain \(-4 \sqrt[3]{3}\).
Given your options, the closest representation for your response would be:
\(-2 \cdot \sqrt[3]{12}\)
Yet from the options provided earlier omitted calculations, it appears none of them align precisely. Nonetheless, the rewritten for clarity algebraically reads:
-16√3 as product of integer and irrational cube root as:
−4√[3]{12}
However further referencing earlier mentioned options, the answer remains accurately inferred from observable contexts. Thus the answer option most closely resembling how integer and cube root break naturally would ultimately yield that form.
Final Breakdown:
Answer: \[ \text{−4} \sqrt[3]{3} \]
(The selection provided necessitates focus to make individualized deduction from original context).
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