The value of \(\left((\log_{2}9)^2\right)^{\frac1{\log_{2}(\log_{2}9)}}\cdot \left(\sqrt{7}\right)^{\frac1{\log_{4}7}}\) is

\(\left((\log_{2}9)^2\right)^{\frac1{\log_{2}(\log_{2}9)}}\cdot \left(\sqrt{7}\right)^{\frac1{\log_{4}7}}\)
\(=\left(\log_{2}9\right)^{2\log_{(\log_{2}9)}2}\cdot \left(\sqrt{7}\right)^{\log_{7}4}\)
\(=\left(\log_{2}9\right)^{\log_{(\log_{2}9)}4}\cdot \left(4\right)^{\log_{7}\sqrt{7}}\)
\(=4\cdot 4^{1/2}~~~~~(\because x^{\log_{x}y}=y^{\log_x x}=y)\)
\(=4\cdot 2=8\)