To simplify the expression \((2^2 \cdot 2^{-6})^{-3} \cdot (2)^0\), let's break down the steps clearly and examine John's and Carol's interpretations.
Step 1: Simplifying Inside the Parentheses
Start with the expression inside the parentheses: \[ 2^2 \cdot 2^{-6}. \] Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents: \[ 2^2 \cdot 2^{-6} = 2^{2 + (-6)} = 2^{-4}. \]
Step 2: Apply the Outer Exponent
Next, we take the result \(2^{-4}\) and raise it to the power of \(-3\): \[ (2^{-4})^{-3}. \] Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we multiply the exponents: \[ 2^{-4 \cdot -3} = 2^{12}. \]
Step 3: Simplifying the Remaining Factor
Now, we have: \[ 2^{12} \cdot 2^0. \] Again, applying the property of exponents: \[ 2^{12} \cdot 2^0 = 2^{12 + 0} = 2^{12}. \]
Final Result
Thus, the fully simplified result is: \[ 2^{12}. \]
Evaluating John's Understanding
John's belief is that the answer was 1 because he thought the expression simplifies to something raised to the zero power. However, he incorrectly interpreted parts of the expression. The step where he applies his zero exponent rule fails because he did not appropriately account for the calculations leading to \(2^{12}\).
Evaluating Carol's Understanding
Carol's reasoning is flawed because although she recognizes the multiplication property of exponents, she miscalculates the powers. Specifically, she mistakenly equates \(2^{-4}\) to \(4^n\) without adequately converting the bases (the correct representation is \(4\) as \(2^2\), but the powers must match correctly). The error lies in her transformation of \(2^{-4}\) to \(4^{12}\) instead of recognizing that \(2^{2 \cdot n}\) allows for a simpler representation of the powers.
Correct Answer
The correct answer, fully simplified, is: \[ 2^{12}. \]
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