To simplify the expression \( (2 - \sqrt{2})^3 \cdot 2^{-\frac{1}{2}} \), we can analyze and simplify each part step by step.
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Understanding the expression:
- We know \( (2 - \sqrt{2})^3 \) is a separate term and we also have \( 2^{-\frac{1}{2}} \), which is the same as \( \frac{1}{\sqrt{2}} \).
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Exponent properties:
- To simplify the expression overall, consider the meaning of \( -\frac{1}{2} \) in the exponent. This term affects how we think about the final answer, as it suggests taking the reciprocal of the square root of 2.
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Potential simplifications:
- \( (2 - \sqrt{2})^3 \) cannot be simplified further without directly calculating.
- The term \( 2^{-\frac{1}{2}} \) becomes \( \frac{1}{\sqrt{2}} \).
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Combining the terms:
- The expression still has \( (2 - \sqrt{2})^3 \cdot \frac{1}{\sqrt{2}} \).
To determine which option matches our simplified expression, none of the potential direct simplifications of \( (2 - \sqrt{2})^3 \) gives us a clean form, but we can notice the nature of \( (2 - \sqrt{2}) \):
- The value \( 2 - \sqrt{2} \) is approximately \( 2 - 1.414 \approx 0.586 \).
- When raised to the third power and then multiplied by \( \frac{1}{\sqrt{2}} \), the numerical complexity might yield a value between integers.
As none of the choices are verified through algebraic details but instead via numerical analysis with radical simplifications, we can conclude the original expression can be checked against basic forms:
- The most reasonable conclusion is:
- \( 2 \) or \( 2^{\frac{1}{2}} \) might especially suggest a relative space where \( \sqrt{2} \) typically matches bounds rather than unusual polynomial forms.
Thus, from the provided options, it appears \( 2 \) would be a more aligned choice to the numeric behavior as compared to radicals without complexity.
Final Answer: \( 2 \)
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