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to powers that are not positive integers. When is non-zero, the definition is compatible with the multiplication rule: . A similar argument suggests the definition

for negative integer powers, and in particular for any nonzero number , and also the definition

for fractional powers (when and are both integers). For example, , meaning , which is the definition of square root: .

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define for any positive real base

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