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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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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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