If you memorize some of the "nice" log10 approximations like ... log10 2 ~= 0.3 log10 2.5 ~= 0.4 log10 3 ~= 0.5 log10 4 ~= 0.6 ... then 250

If you memorize some of the "nice" log10 approximations like ... log10 2 ~= 0.3 log10 2.5 ~= 0.4 log10 3 ~= 0.5 log10 4 ~= 0.6 ... then 25000^(1/8) could be done mentally as log10 of 25000 is 4.4 4.4 / 8 is 0.55 antilog10 of 0.55 is approximately 3.5