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**Unformatted text preview: **The proof of Theorem 5.6 is based on the deﬁnition of the principal roots and properties of expo√
nents. To establish the product rule, consider the following. If n is odd, then by deﬁnition n xy
√√n
√n√n
√n
is the unique real number such that ( n xy ) = xy . Given that n x n y = ( n x) n y = xy ,
√√
√
√
it must be the case that n xy = n x n y . If n is even, then n xy is the unique non-negative real
√
√n
√
number such that ( n xy ) = xy . Also note that since n is even, n x and n y are also non-negative
√√
√√
√
and hence so is n x n y . Proceeding as above, we ﬁnd that n xy = n x n y . The quotient rule is
proved similarly and is left as an exercise. The power rule results from repeated application of the
√
product rule, so long as n x is a real number to start with.2 The last property is an application of
the power rule when n is odd, and the occurrence of the absolute value √
when n is even is due to
√
4
n
4 = 4 16 = 2 = | − 2|, not −2.
the requirement that x ≥ 0 in Deﬁnition 5.4. For instance, (−2)
It’s this la...

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