<s>[INST] How would you explain to a 9th grader the negative exponent rule? Let us assume that the students haven't been exposed to these two rules: $a^{x+y} = a^{x}a^{y}$ and $\frac{a^x}{a^y} = a^{x-y}$. They have just been introduced to the generalization: $a^{-x} = \frac{1}{a^x}$ from the pattern method: $2^2 = 4, 2^1 = 2, 2^0 = 1, 2^{-1} = \frac{1}{2}$ etc. However, some students confuse $2^{-3}$ to be $(-2)(-2)(-2)$ since they are familiar with $2^{3} = 2 \cdot 2 \cdot 2$. This is a low-income urban school and most kids in this algebra class struggle with math dealing with exponents, fractions and decimals. What would be the best approach to reach all 32 students? 
 [/INST] 
some students confuse $2^{-3}$ to be $(-2)(-2)(-2)$

Simple, $(-2)(-2)(-2)$ means ${-2}^{3}$, that is the first thing you have to show when they make that mistake.
Edit: That is the number minus 2, not a 2 combined with the unarry (-) and a exponent. Or it probably is, you might want to write ${(-2)}^{3}$ if it confuses you, i'm not even sure which bind stronger.
Next, show the relation between $2^3$ and $2^2$. One equals $2\cdot2\cdot2$ and the other $2\cdot2$. So $2^3$ = $2^2\cdot2$ and $\frac{2^3}{2} = 2^2$. Also, ${2\cdot2\cdot2\over2}=2\cdot2$
We can then move forwards (well, backwards),
$\frac{2^2}{2} = 2^1$,
$\frac{2^1}{2} = 2^0$,
$\frac{2^0}{2} = 2^{-1}$
etc.
</s>