$\large 3^{\log_{3}^{2}x}+x^{\log_{3}x}=162$
=> $\large (3^{\log_{3}x})^{\log_{3}x}+x^{\log_{3}x}=162$
=>$\large (x)^{\log_{3}x}+x^{\log_{3}x}=162$ [ $\large 3^{\log_{3}x}=x$]
=>$\large 2x^{\log_{3}x}=162$
=>$\large x^{\log_{3}x}=81$
taking log both the side with base 3,
=>$\large \log_{3}^{2}x=\log_{3}81$
=>$\large \log_{3}^{2}x=4$
=>$\large \log_{3}x=\pm 2$
so $\large x=9$ or $\large x=\frac{1}{9}$