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Algebra Level 3

How many digits does 8 1000 8^{1000} have?


The answer is 904.

Jack Sacks by Jack Sacks , 18, United Kingdom

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2 solutions

Lee Care Gene
Feb 15, 2016

8 1000 = 1230231922161117176931558813276752514640713895736833715766118029160058800614672948775360067838593459582429649254051804908512884180898236823585082482065348331234959350355845017413023320111360666922624728239756880416434478315693675013413090757208690376793296658810662941824493488451726505303712916005346747908623702673480919353936813105736620402352744776903840477883651100322409301983488363802930540482487909763484098253940728685132044408863734754271212592471778643949486688511721051561970432780747454823776808464180697103083861812184348565522740195796682622205511845512080552010310050255801589349645928001133745474220715013683413907542779063759833876101354235184245096670042160720629411581502371248008430447184842098610320580417992206662247328722122088513643683907670360209162653670641130936997002170500675501374723998766005827579300723253474890612250135171889174899079911291512399773872178519018229989376 8^{ 1000 }=123023192216111717693155881327675251464071389573683 371576611802916005880061467294877536006783859345958 242964925405180490851288418089823682358508248206534 833123495935035584501741302332011136066692262472823 975688041643447831569367501341309075720869037679329 665881066294182449348845172650530371291600534674790 862370267348091935393681310573662040235274477690384 047788365110032240930198348836380293054048248790976 348409825394072868513204440886373475427121259247177 864394948668851172105156197043278074745482377680846 418069710308386181218434856552274019579668262220551 184551208055201031005025580158934964592800113374547 422071501368341390754277906375983387610135423518424 509667004216072062941158150237124800843044718484209 861032058041799220666224732872212208851364368390767 036020916265367064113093699700217050067550137472399 876600582757930072325347489061225013517188917489907 9911291512399773872178519018229989376

Therefore 8 1000 8^{ 1000 } has 904 904 digits.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Come on...this is not a practical approach...use logarithms instead

Mohit Gupta - 5 years, 4 months ago

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I know. I'm just showing the unpractical approach to this question.

Lee Care Gene - 5 years, 4 months ago

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K...... i presume it was for fun :p

Mohit Gupta - 5 years, 4 months ago

⌊ 3000 log ⁡ 2 ⌋ = 904 \lfloor 3000\log 2 \rfloor=904

Akshat Sharda - 5 years, 4 months ago

ha ha ha ha, Nice solution,hahaha

Guillermo Templado - 5 years, 3 months ago

x = 8 1000 x = 8^{1000}

l o g x = l o g 8 1000 = 1000 l o g 8 = 1000 ∗ 0.903 = 903 log x = log 8^{1000} = 1000 log 8 = 1000 * 0.903 = 903

So, total no of digits = 903 + 1 = 904

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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