To simplify the expression \( 3 - \sqrt{216}/27^{1/3} \), we can break it down step by step.
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Calculate \( 27^{1/3} \): \[ 27 = 3^3 \quad \text{so} \quad 27^{1/3} = (3^3)^{1/3} = 3^{3 \cdot \frac{1}{3}} = 3^1 = 3 \]
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Calculate \( \sqrt{216} \): We can express \( 216 \) as a product of prime factors: \[ 216 = 2^3 \cdot 3^3 \] Therefore, \[ \sqrt{216} = \sqrt{2^3 \cdot 3^3} = \sqrt{2^3} \cdot \sqrt{3^3} = 2^{3/2} \cdot 3^{3/2} = 2\sqrt{2} \cdot 3\sqrt{3} = 6\sqrt{6} \]
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Substitute back into the expression: \[ 3 - \frac{\sqrt{216}}{27^{1/3}} = 3 - \frac{6\sqrt{6}}{3} \] \[ = 3 - 2\sqrt{6} \]
Since this does not correspond to the provided options (18, 8, 2, 1/2), it appears that we had a miscalculation either in rewriting the expression or possibly in the interpretation of the question.
To address the original expression more directly:
Given: \[ x - \frac{\sqrt{216}}{27^{1/3}} \] Express everything in terms of base \( 3 \):
\(\sqrt{216} = 6\sqrt{6}\) and \(27^{1/3} = 3\). Thus:
\[ = 3 - 2\sqrt{6} \]
However, we directly see that it creates issues, as no integer or simple fraction emerges. My assumption would be a recheck had you initially provided it correctly. If it's calculated directly as shown; none of the answer choices seem to align.
Therefore, please check the original details to ensure the values presented are correct. There might be a need for elaboration or re-evaluation based on \(-\sqrt{}\) or behavior in the context of roots and fractions not provided directly within integer bounds or direct acceptance.