Question

The value of ${\left(\right({\log }_{2}9{)}^2)}^{\frac{1}{{\log }_2{\log }_{2}9}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is ________?

Answer 1

${\left(\right({\log }_{2}9{)}^2)}^{\frac{1}{{\log }_2{\log }_{2}9}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$

$={\log }_2{9}^{\frac{2}{{\log }_2\left({\log }_{2}9\right)}}\times 7^{\frac{1/2}{{\log }_{4}7}}$

$={\left({\log }_{2}9\right)}^{2{\log }_{{\log }_{2}9}2}\times 7^{\frac{1}{2}{\log }_{7}4}$

$=4\times 2=8$.