Question

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

Answer 1

$(\sqrt{7}{)}^{\frac{1}{{\log }_{4}7}}=(\sqrt{7}{)}^{{\log }_{7}4}=(7{)}^{\frac{1}{2}{\log }_{7}4}=(4{)}^{\frac{1}{2}}=2$

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

${\left({\left({\log }_{2}9\right)}^2\right)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}=2\times 4=8$