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
I bolded the important parts to the problem, Please help :)
+26367 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 \(m + n\)
\(\begin{array}{|rcll|} \hline \mathbf{a^b} &=& \mathbf{b^a} \quad | \quad \mathbf{b = 9a} \\\\ a^{9a} &=& (9a)^a \\\\ a^{9a*\frac{1}{a}} &=& \left(9a\right)^{a*\frac{1}{a}} \\\\ a^{9} &=& 9a \quad | \quad *(a^{-1}) \\\\ a^{9}a^{-1} &=& 9a^1a^{-1} \\\\ a^{9-1} &=& 9a^{1-1} \\\\ a^{8} &=& 9a^0 \quad | \quad a^0 = 1 \\\\ a^{8} &=& 9 \\\\ a^{8*\frac{1}{8}} &=& 9^{\frac{1}{8}} \\\\ a &=& 9^{\frac{1}{8}} \\\\ a &=& (3^2)^{\frac{1}{8}} \\\\ a &=& 3^{2*\frac{1}{8}} \\\\ a &=& 3^{\frac{2}{8}} \\\\ a &=& 3^{\frac{1}{4}} \\\\ \mathbf{a} &=& \mathbf{\sqrt[4]{3}} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline m+n &=& 3+4 \\ \mathbf{m+n} &=& \mathbf{7} \\ \hline \end{array}\)
