Question

Natalie tried to evaluate the expression
\[\left( 4^{-3} \cdot 2^{-3} \right)^{0}\].

\[\begin{aligned}
&\phantom{{}={}}\left( 4^{-3} \cdot 2^{-3} \right)^{0}
\\\\
&=\left( 8^{-3}\right)^{0} &\text{Step } 1
\\\\
&= 8^{0} &\text{Step } 2
\\\\
&=0&\text{Step } 3
\end{aligned}\]
Did Natalie make a mistake? If so, in which step?
Choose 1 answer:
Choose 1 answer:
(Choice A) Natalie did not make a mistake.
A
Natalie did not make a mistake.
(Choice B) Natalie made a mistake in Step
\[1\].
B
Natalie made a mistake in Step
\[1\].
(Choice C) Natalie made a mistake in Step
\[2\]
C
Natalie made a mistake in Step
\[2\]
(Choice D) Natalie made a mistake in Step
\[3\].
D
Natalie made a mistake in Step
\[3\].

Answer 1

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.**