Clock Diet

at first , $7^2 +24^2$ = $625$

now $\sqrt (625)$ =25

and , $\sqrt 25$ =5

Recall that $7, 24, 25$ is one of the Pythagorean triples.

$\displaystyle \sqrt{7^2 + 24^2} = 25$

$\displaystyle \sqrt{\sqrt{7^2 + 24^2}} = \sqrt{25} = \boxed{5}$

$\sqrt{\sqrt{7^2+24^2}} = (7^2+24^2)^{1/4} = \sqrt[4]{625} = \boxed{5}$