Natalie is trying to evaluate the expression $(4^{-3} \cdot 2^{-3})^0$. The steps taken are shown and we need to determine if a mistake was made and in which step.

1. Problem Description

Natalie is trying to evaluate the expression $(4^{-3} \cdot 2^{-3})^0$. The steps taken are shown and we need to determine if a mistake was made and in which step.

2. Solution Steps

The initial expression is $(4^{-3} \cdot 2^{-3})^0$.

Step 1: $(4^{-3} \cdot 2^{-3})^0 = (8^{-3})^0$

This step combines $4^{-3}$ and $2^{-3}$ as $8^{-3}$.

We know that $4^{-3} \cdot 2^{-3} = (4 \cdot 2)^{-3} = 8^{-3}$.

So, Step 1 is correct.

Step 2: $(8^{-3})^0 = 8^0$

In this step, $(8^{-3})^0$ is simplified to $8^0$. However, the exponent rule $(a^m)^n = a^{m \cdot n}$ should be applied here. So, $(8^{-3})^0 = 8^{-3 \cdot 0} = 8^0$.

So, Step 2 is correct.

Step 3: $8^0 = 0$

Here, any non-zero number raised to the power of 0 is equal to 1, not

0. Therefore, $8^0 = 1$, not

0. This is where the mistake occurred.

$a^0 = 1$ if $a \ne 0$.

3. Final Answer

Natalie made a mistake in Step 3.