To evaluate the expression
\[ \left( 4^{-3} \cdot 2^{-3} \right)^{0}, \]
we need to analyze each of Natalie's steps.
Step 1:
Natalie claims
\[ \left( 4^{-3} \cdot 2^{-3} \right)^{0} = \left( 8^{-3}\right)^{0}. \]
Let's verify this:
- We can express \(4\) as \(2^2\), so \(4^{-3} = (2^2)^{-3} = 2^{-6}\).
- Therefore, \(4^{-3} \cdot 2^{-3} = 2^{-6} \cdot 2^{-3} = 2^{-6 - 3} = 2^{-9}\).
- Now we can re-write the expression as \((2^{-9})^{0}\).
Since \(4^{-3} \cdot 2^{-3}\) does not equal \(8^{-3}\), this means that Natalie made a mistake in Step 1.
Step 2:
Natalie claims that
\[ \left( 8^{-3} \right)^{0} = 8^{0}. \]
This conversion is correct regardless of the value, as any non-zero number raised to the power of \(0\) is \(1\).
Step 3:
Natalie claims that
\[ 8^{0} = 0. \]
This is incorrect. \(8^{0} = 1\).
Conclusion:
Natalie makes a mistake in Step 1 when she incorrectly simplifies \(4^{-3} \cdot 2^{-3}\) to \(8^{-3}\), and she makes another mistake in Step 3 stating that \(8^{0} = 0\) instead of \(1\).
However, since the question specifically asks for where she made a mistake that directly affects her evaluations, the best choice would be to identify the mistake that led to an incorrect result.
Thus, the correct choice is:
(Choice B) Natalie made a mistake in Step 1.
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