Negative Exponents and Multiplicative Inverses
1. Extending Exponents Below Zero
For any non-zero base $a$ and any positive integer $m$:
$$a^{-m} = \frac{1}{a^m}\quad \text{and}\quad \frac{1}{a^{-m}} = a^m$$
- $a^{-m}$ is the multiplicative inverse of $a^m$ because:
$$a^m \times a^{-m} = a^{m + (-m)} = a^0 = 1$$
THE EXPONENTIAL NUMBER LINE
... 10^(-3) 10^(-2) 10^(-1) 10^0 10^1 10^2 10^3 ...
1/1000 1/100 1/10 1 10 100 1000
<-------------------------- Divide by 10 | Multiply by 10 ------------------------>
2. Reciprocals of Fractions Under Negative Exponents
For any rational fraction $\frac{a}{b}$ ($a, b \neq 0$):
$$\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m$$
- Example: $\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$
