Find the value of $\left((\log_29)^2\right)^{1/\log_2(\log_29) }\times \left(\sqrt{7}\right)^{1/\log_47}$ The value of$\left((\log_29)^2\right)^{1/\log_2(\log_29) }\times \left(\sqrt{7}\right)^{1/\log_47}$.
My approach :
I am able to solve this part : $(\sqrt{7})^{1/\log_47}$ by changing base :
$(\sqrt{7})^{1/\log_47}   =  (7)^{2\log_74} = 16$
But I am unable to figure how to solve this part :
$\Bigl((\log_29)^2\Bigr)^{1/\log_2(\log_29)}  $
Please help on this … thanks

Your first part is incorrect. $$(\sqrt{7})^{1/\log_47}   =  (7^{1/2})^{\log_74} = 2$$
Use @00xxqhxx00's hint to solve for the 2nd part.  Let $x = \log_29$ $$\Bigl((\log_29)^2\Bigr)^{1/\log_2(\log_29)} = (x^2)^{1/\log_2 x} = (x^2)^{\log_x 2} = 4$$
Hence the required product is $2 \times 4 = 8$.
