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Let \(a\) and \(b\) be positive real numbers such that \(a^b=b^a\) and \(b=9a.\) Then \(a\) can be expressed in the form \(\sqrt[m]{n},\) where \(m\) and \(n\) are positive integers, and \(n\) is as small as possible. Find \(m+n.\)

 Apr 1, 2019

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 #1
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\(a^{9a}=(9a)^a\\ a^9=9a\\ a^8 = 9\\ a = 9^{1/8} = 3^{1/4} = \sqrt[4]{3}\\ 4+3=7\)

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 Apr 1, 2019
 #1
avatar+6192 
+2
Best Answer

\(a^{9a}=(9a)^a\\ a^9=9a\\ a^8 = 9\\ a = 9^{1/8} = 3^{1/4} = \sqrt[4]{3}\\ 4+3=7\)

Rom Apr 1, 2019

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