70843029071054204781573025995798769907739696815288329699328

has 59 digits.

We are told we are using a "simple" calculator without a definition of "simple" but:

If the calculator is able to use 13-digit numbers use the fact that 59 = 2*23+13 (regarding the number of digits in the given number). So take the 23rd root of the first 13 digits and use the beginning of the result scaled up to a 3-digit number:

7084302907105 ^ (1/23) = 3.61999999999999

so the answer is 362. As a check, raising 362 to the 23rd power gives  7.08430290710542 x 10^58, if the simple calculator has the scientific notation capabilities.

Or, if your calculator has a log key for common logs, take log(7.0843029) and add 58; divide by 23; then take the antilog of that sum:

log(7.0843029) + 58 = 58.8502971218272

/ 23 = 2.55870857051423

inv log = 361.999999984214 (however that's done on your calculator)

giving 362.

This second method has the advantage of not worrying about how many digits of accuracy you have, or knowing beforehand how many digits are to be in the answer.

Not relevant to the solving of this puzzle, but as for being sure that all the original digits were correct, extended precision gives:

>> sym(362)^23

ans =

70843029071054204781573025995798769907739696815288329699328

or

>> a=strrep('70,843,029,071,054,204,781,573,025,995,798,769,907,739,696,815,288,329,699,328',',','')

a =

    '70843029071054204781573025995798769907739696815288329699328'

>> a=sym(a)

a =

70843029071054204781573025995798769907739696815288329699328

>> a^(1/23)

ans =

362