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Let a and b be positive real numbers such that a^b = b^a, b = 9a, Then can be expressed in the form sqrt(n,m) where m and are positive integers

and n is as small as possible. Find m + n

Apr 6, 2020

#1
+24972
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Let $$a$$ and $$b$$ be positive real numbers such that $$a^b = b^a,\ b = 9a$$,
Then $$a$$ can be expressed in the form $$sqrt(n,m)$$ where $$m$$ and $$n$$ are positive integers
and $$n$$ is as small as possible.

Find $$m + n$$

$$\begin{array}{|rcll|} \hline \mathbf{a^b} &=& \mathbf{b^a} \quad | \quad \mathbf{b = 9a} \\\\ a^{9a} &=& (9a)^a \\\\ a^{9a*\frac{1}{a}} &=& \left(9a\right)^{a*\frac{1}{a}} \\\\ a^{9} &=& 9a \quad | \quad *(a^{-1}) \\\\ a^{9}a^{-1} &=& 9a^1a^{-1} \\\\ a^{9-1} &=& 9a^{1-1} \\\\ a^{8} &=& 9a^0 \quad | \quad a^0 = 1 \\\\ a^{8} &=& 9 \\\\ a^{8*\frac{1}{8}} &=& 9^{\frac{1}{8}} \\\\ a &=& 9^{\frac{1}{8}} \\\\ a &=& (3^2)^{\frac{1}{8}} \\\\ a &=& 3^{2*\frac{1}{8}} \\\\ a &=& 3^{\frac{2}{8}} \\\\ a &=& 3^{\frac{1}{4}} \\\\ \mathbf{a} &=& \mathbf{\sqrt[4]{3}} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline m+n &=& 3+4 \\ \mathbf{m+n} &=& \mathbf{7} \\ \hline \end{array}$$

Apr 6, 2020
#2
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Thank you so much for explaining this to me!

Apr 6, 2020