Can Modern Calculators Solve (8^2003 + 6^2003) / 49?

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The discussion centers on the mathematical expression (8^2003 + 6^2003) / 49, which proves too complex for standard calculators. Participants explore logarithmic approximations, determining that 8^2003 dominates the sum, allowing for simplification. The approximate value is calculated as 1.5816 * 10^1807, with a detailed decimal expansion provided. The exact answer is significantly lengthy, showcasing the intricacies of large number calculations. The conversation highlights the challenges of handling such large exponentials and the limitations of typical computational tools.
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Even calculator can't solve this...

(8^2003+6^2003)/49=?

Who can solve this?
 
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8^{2003} = 10^{2003*log_{10}8} = 10^{1808.88924} = 7.75 * 10^{1808}

6^{2003} = 10^{2003*log_{10}6} = 10^{1558.63695} = 4.335 * 10^{1558}

The second can be neglected in comparison to the first giving

(8^{2003}+6^{2003})/49 = (about)~ 1.5816 * 10^{1807}

Also, if you wrote out the full number (in decimal expansion) in all its gory detail, the digits after the decimal point would be .04081632653... for whatever that's worth.
 
Last edited:
Well, I didn't do the arithmetic, but the answer is

110699562591395625875266564924664097310974709512929097789175911617601046454386434533363625521484979625995866415556215005295416133194216915274253564533823808741077943952672510750053902252973351456954446065237629947590350789140326338000369452446840290082144467142045178897824353048670587312080224265459784206322855624222209627723142215253195344567886035909787077726154777006320959340849530325395879878645900740268181333554850414711304850535157904078675787256343935250506945484632580361260539052711292135393964305721326164034185323257036401562064939229517300512382832378049844366899876452596800955806420515203883818227428415114965787363900634482730394595855901269536477229409114070793987701830131336961091311521917948013656268286590558122942173068042452683584041597393131362115133639708695813827166004500777279577036282790847755599032408449092996078742277433061567299573578686181516485959696804800782360503486653196727233559342142902572693633989299389429306974042715177724597042437294815094190562672889102497380940799682008786172658233122667320788812823727542218978920706958595020667644482284910188722257885885010942161301492075556745333658141372608295823959953020682907150950904979738403629013487885794876482903052545000231653116968614767872331012877153046697429893686056710998765089687784315909815005258973458134465324287048736599893117632184261219444205283823063443143881361385024937369090408673392508695811724593424721886293429928788520837043197212862355297138917320309016807737396766789530004574754173260914474719164345351785481277844433138044222960404781867964113414476335964916396492692453035888217537736329111435482807135250055792650863889863243772287640514370785654038798197610745561837609318531738132668887103225077809387770418238590052166821081931085756853573798577748014227700555866270649651879215104

divided by 7, which can be shortened to approximately

1.5814223227342232267895223637314066 * 10^1807

Mathematica did it, not me :redface: Excuse the strange spacing too...it's late and I feel no urge to write simple latex.
 
Last edited:
The exact answer is:

15814223227342232267895223560666299615853529930418442541310844516800149493483776361909089360212139946570838059365173572185059447599173845039179080647689115534439706278953215821436271750424764493850635152176804278227192969877189476857195636063834327154592066734577882699689193292667226758868603466494254886617550803460315661103306030750456477795412290844255296818022111000902994191549932903627982839806557248609740190507835773530186407219308272011239398179477705035786706497804654337322934150387327447913423472245903737719169331893862343080294991318502471501768976054007120623842839493228114422258060073600554831175346916444995112480557233497532913513693700181362353889915587724399141100261447333851584473074559706859093752612370079731848881866863207526226291656770447337445019091386956544832452286357253897082433754684406822228433201207013285154106039633294509614224796955168788069422813829257254622929069521885246747651334591843224670519141328484204186710577530739674942434633899259299170080381841300356768705828526001255167522604731809617255544689103934602711274386708370717238234926040701455531751126555001563165900213153650963619094020196086899403422850431526129592992986425676914804144783983684982354700436077857175950445281230681124618716125307578099632841955150958714109298526826330844259286465567636876352189183864105228556159661740608745634886469117580491877697337340717848195584344096198929813687389227632103126613347132684074405291885316123193613876988188615573829676770966684218572082107739037273496388452049335969354468263490448292031851486397409709159059210905137845199498956064719412602505390904158776497543876464293684664409127123320538898234359195826522005542599658677937405372759790248304666983871889296829912538631176941436023831583133012250979081971225392573461100079409467235664554173586 + 2/7

As 82003+62003 is:

774896938139769381126865954472648681176822966590503684524231381323207325180705041733545378650394857381971064908893505037067912932359518406919774951736766661187545607668707575250377315770813460198681122456663409633132455523982284366002586167127882030575011269994316252284770471340694111184561569858218489444259989369555467394061995506772367411975202251368509544083083439044246715385946712277771159150521305181877269334883952902979133953746105328550730510794407546753548618392428062528823773368979044947757750140049283148239297262799254810934454574606621103586679826646348910568299135168177606690644943606427186727591998905804760511547304441379112762170991308886755340605863798495557913912810919358727639180653425636095593878006133906860595211476297168785088291181751919534805935477960870696790162031505440957039253979535934289193226859143650972551195942031430971097015050803270615401717877633605476523524406572377090634915395000318008855437925095726005148818299006244072179297061063705659333938710223717481666585597774061503208607631858671245521689766092795532852444948710165144673511375994371321055805201195076595129110444528897217335606989608258070767719671144780350056656334858168825403094415200564135380321367815001621571818780303375106317090140071326882009255802396976991355627814490211368705036812814206941257270009341156199251823425289828536109436986761444102007169529695174561583632860713747560870682072153973053204054009501519645859302380490036487079972421242163117654161777367526710032023279212826401323034150417462498368944911031966309560722833473075748793901334351754414775448847171251217522764154303780048379649946750390548556047229042706406013483600595499578271587383275218932863265229722166928682209722575544665714392927670130365167747573517600297975016590044236099593903891063894547563154505728

Edit my fault[/size] I misread the post, I've corrected it now
 
Last edited:
The question is...

Why?
 
FZ+ said:
The question is...

Why?
The answer is,...

Why not?
 
Yes, that is the question.

EDIT : DAMN, beat to the punch...and by a dissenter, at that. bah !
 

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