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$\large \begin{aligned} x & = 81^\frac 1{\log_5 3} + 27^{\log_9 36} + 3^\frac 4{\log_7 9} \\ & = 3^\frac {4\log_3 5}{\log_3 3} + 3^\frac {3\log_3 36}{\log_3 9} + 3^\frac {4 \log_3 7}{\log_3 9} \\ & = 3^{4\log_3 5} + 3^\frac {6\log_3 6}2 + 3^\frac {4 \log_3 7}2 \\ & = 3^{4\log_3 5} + 3^{3\log_3 6} + 3^{2 \log_3 7} \\ & = 3^{\log_3 5^4} + 3^{\log_3 6^3} + 3^{\log_3 7^2} \\ & = 5^4 + 6^3 + 7^2 \\ & = 625 + 216 + 49 \\ & = \boxed{890} \end{aligned}$

$\Large \begin{aligned} 81^{\frac 1{\log_5 3}} + 27^{\log_9 36} + 3^{\frac 4{\log_7 9}} &= 81^{\log_3 5} + 27^{\log_9 36} + 3^{4{\log_9 7}} \\ &= 3^{4{\log_3 5}} + 3^{3\log_{3^2} 36} + 3^{4{\log_{3^2} 7}} \\ &= 3^{\log_3 5^4} + 3^{\log_3 {36}^{{3}/{2}}} + 3^{\log_3 7^2} \\ &= 5^4 + 36^{{3}/{2}} + 7^2 \\ &= 625 + 216 + 49 \\ &= \boxed{890} \end{aligned}$