If $\;\log_m mn + \log_n mn = 49$, find $(\log_m n)^{3/4} + (\log_n m)^{3/4}.$ Given that $\log_m mn + \log_n mn = 49,\;$ find the value of $$(\log_m n)^{3/4} + (\log_n m)^{3/4}.$$
I have rewritten this expression but have hit a dead end. 

Let $\sqrt[4]{\log_nm}=x$.  Hence, $x>0$ and  $$ x^4+\frac{1}{x^4}=47 \iff
\left(x^2+\frac{1}{x^2}\right)^2=49$$ 
$$\iff x^2+\frac{1}{x^2}=7$$ 
$$\iff\left(x+\frac{1}{x}\right)^2=9$$ 
$$\iff x+\frac{1}{x}=3.$$
Thus, $$\left(\log_m n\right)^{3/4} + \left(\log_n m\right)^{3/4}=x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)^3-3\left(x+\frac{1}{x}\right)=27-9=18.$$
