Derive an approximate value of
N=301^300.
Present your answer as A*10^b (b being an integer number), plus estimated accuracy of C%.
-
The more accurate the better !
Two versions: calculator and in-the-head:
Calculator:
log(301) = 2.478566495593843
Multiplying this by 300 gives
743.569948678153
The value of b will be 743; A will be 10^.569948678153 ~= 3.714913262884367.
However, as the mantissa (fractional part) of 743.569948678153 (the common log of the answer) has only 12 digits, the answer is good only to 12 significant figures: 3.71491326288. This implies a possible error of 0.000000000005. Divide this by 3.71491326288 and multiply by 100 and you get (giving in reciprocal form) about one seven-billionth of 1 percent for C.
Final estimate: 3.71491326288 * 10^743.
In the head:
Memorized log of 3 is 0.48. Log of 301 is not much more than 2.48. (Actually, as one saw, it's less, due to inaccuracy of 0.48)
Multiplying that by 300 gives 744. The answer is about 10^744, and is good only as an order of magnitude (i.e., A is taken as 1).
The more accurate the better:
UBASIC:
3714913262884428002203860381042753061246398291073460798527283925882177161876753714436540778237002119
6039399005538681942709905553351570713271550903363119430119536650553165800367498425601686951660367727
1210694973161302379215546618121941875503526328409399773985842487531777424870446301434188776181804165
2529773499563950877524733403212781858894141911623335112230175615003131038072635399054944663771667246
8706763640855760954926786685185377863036225012672726708051363860893721150992205882621662058493084477
2504318568381244175338656159795280605057300322622067983143277751388577725127925961814964359688294383
9368040416069311045289735823486253941594451527494082480438580446517512768413160294549673169313334924
00734983527772452768213625149099609236590001
from
10 point 222
20 print 301^300
check on the last few digits:
36590001 done
from
p = 1
For i = 1 To 300
p = p * 301
q = Int(p / 100000000)
p = p - q * 100000000#
Next
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Posted by Charlie
on 2019-07-17 15:46:34 |
answers
