To simplify the expression \( (2^{-\sqrt{}})^3 \cdot 2^{-12} \), we can apply the properties of exponents.
-
Start by simplifying the first part: \[ (2^{-\sqrt{}})^3 = 2^{-3\sqrt{}} \]
-
Now, combine it with the second part: \[ 2^{-3\sqrt{}} \cdot 2^{-12} = 2^{-3\sqrt{} - 12} \]
Now, we can analyze the options provided:
- The expression simplifies down to a single exponential base. As such, none of the options directly correspond without knowing the value of \(\sqrt{}\).
However, if we consider particular values like \( \sqrt{2} \), it results in a different form.
To conclude, the expression simplifies to \( 2^{-3\sqrt{} - 12} \). Without specific numeric options tied to \(\sqrt{}\), the question might benefit from specific numerical interventions.
If you choose based on the strictly available numeric options without additional context or checking values, the expression cannot particularly resolve to \(2\), \(\sqrt{2}\), \(2^2\), or \(2^{12}\) as listed options without additional context confirmation.
Based solely on the exponent rules applied,** the exact resolution cannot conclude an exclusive alternate equivalent given. Further definitions for \(\sqrt{}\) will clarify an option link resolutely.