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Negative Exponents

What should $ a^{-1}$ be? Multiplying it by $ a$ gives, using property (1),

$\displaystyle a^{-1} \cdot a = a^{-1} a^1 = a^{-1+1} = a^0 = 1. $

Dividing through by $ a$ then gives

$\displaystyle \zbox {a^{-1} = \frac{1}{a}.} $

Similarly, we obtain

$\displaystyle \zbox {a^{-M} = \frac{1}{a^M}} $

for all integer values of $ M$, i.e., $ \forall M\in{\bf Z}$.

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The Exponent Zero