<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] 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.
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