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# help

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

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Apr 1, 2019

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