Stitz-Zeager_College_Algebra_e-book

# The rst base 10 is often called the common base the

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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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