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? 

Multiplication of exponents:
$4^2 \times 4^3 = 4^5$
The powers are added: $2 + 3 = 5$
Division of exponents:
$\dfrac{4^2}{4^3}\ = 4^{-1}$
The powers are subtracted: $2 - 3 = -1$
Now tell them to try and find the value of $4^{-1}$
Since $4^2 = 16$, and $4^3 = 64$
$\dfrac{16}{64}\  = 1/4 = 4^{-1}$
This is the way I would have wanted to have been taught it.
