Explain how you can use these powers to define 3^( sqrt of 2 )

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Discussion Overview

The discussion revolves around evaluating powers with rational and irrational exponents, specifically focusing on how to define \(3^{\sqrt{2}}\) using rational approximations. Participants also explore a related problem involving the calculation of \(e\) using the expression \((1 + 1/n)^n\) and the determination of an appropriate value for \(n\).

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant shares a series of calculations for powers of 3 with rational exponents, suggesting that \(3^{\sqrt{2}}\) can be approximated by observing the values of \(3^{1414213/1000000}\) and noting that \(3^{\sqrt{2}}\) is slightly larger.
  • Another participant proposes using logarithms to find the value of \(3^{\sqrt{2}}\) by adding logarithmic values, although they express confusion about the process.
  • In a separate problem, a participant attempts to find \(n\) such that \((1 + 1/n)^n\) approximates \(e\) to nine decimal places, using logarithmic manipulation but encounters difficulties with undefined results.
  • Another participant suggests using a calculator to find \(n\) and emphasizes the importance of using sufficient decimal places to ensure accuracy.
  • A later post provides a lengthy numerical output from Maple, presumably as an approximation for \(e\), but does not clarify how this relates to finding \(n\).

Areas of Agreement / Disagreement

The discussion contains multiple competing views on how to approach the problems, with no consensus reached on the methods for defining \(3^{\sqrt{2}}\) or determining \(n\) for the approximation of \(e\).

Contextual Notes

Participants express confusion over the application of logarithmic properties and the implications of using rational approximations for irrational exponents. There are also unresolved issues regarding the calculations and the definitions used in the logarithmic expressions.

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Hey... requested my professor to give me a few word problems to enable me to improve my skills... am having problems solving them however and was hoping that you could guys could help me:

Q1) Use a calculator to evaluate the following powers. Round the results to five decimal placeS. Each of these powers has a rational exponent. Explain how you can use these powers to define 3^( sqrt of 2 ) which has an irrational exponent.

3^(14/10) = 4.65554
3^(141/100)=4.70697
3^(1414/10000)= 4.72770
3^(14142/10000)= 4.72873
3^(141421/100000)= 4.72878
3^(1414213/1000000)= 4.72880


3^( sqrt of 2 ) = 4.728804

so basically the value of 3^(sqrt of 2 ) comes after 3^(1414213/1000000)... hence, you could find the log of 3^14/10 which is .6679697566, then probably do the following : 3^(.667..+.667...+.667...+.667.. +.667 +.667.. +.667).

EDIT: damn .. what i did above doesn't make sense.. I am soo confused.! i do know that i can somehow use logs by finding the log of one value and then adding the solution multiple times to find the value of 3^ ( sqrt of 2 )

_________________________________________________________

Question 2:

Find a value of n for which (1+1/n)^n gives the value of e correct to 9 decimal places. Explain the process you used to find your answer.

log 2.718281828 = log ( 1+1/n)^n
log 2.718281828= n log ( 1+1/n)
log 2.718281828/ log (1+1/n ) = n

would that be the final answer?

thanks
 
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DO NOT DOUBLE POST!For your second problem,u could use a calculator which would give u 10 decimals of "e" (incidentally those are (2.)7182818284(59045...)) (why 10?If you write only 9,u'll never be sure if your calculator will not approximate the 9-th to a unit too much;...79... could be approximated to ...8...) and then use the calculator to give "n" and then find that "n" for which that "8" (the 9-th correct decimal) would appear.

As for alternative solutions,they involve approximations,e.g. the one with the natural logarithm that u found...

Daniel.
 
DO NOT DOUBLE POST!

imo, i cannot delete the previous topic that i made by mistake and hence need a mod to delete it... sorry for double posting.. didnt do it intentionally..

then use the calculator to give "n"

how? even if i do try using 10 decimals of "e", how do i find the value of n??

if i divide log "e" by log (1+1/n) i don't get anything... it comes up as being undefined..
 
Plug 10000.Here's what Maple had to say...2\,71814\,59268\,25224\,86403\,76646\,74913\,14653\,61138\,22649\,22072\,08183\,70865\,87378\,74197\,73771\,51396\,84730\,52781\,47783\,95201\,39855\,02523\,32514\,27278\,51252\,60669\,48976\,10003\,45427\,31187\,99109\,30676\,84981\,85665\,58206\,31434\,30483\,15259\,13526\,35090\,09396\,74095\,16117\,83827\,54701\,95160\,84152\,56234\,06623\,52454\,30366\,88228\,24462\,79113\,55300\,03628\,76646\,77326\,89431\,89206\,81691\,97525\,60437\,44118\,02619\,08090\,91242\,59954\,30139\,97331\,08900\,22516\,98440\,41752\,78311\,74265\,78254\,09053\,90869\,80995\,48471\,29021\,21665\,66588\,14625\,02271\,09169\,30414\,86593\,95043\,47113\,79360\,52635\,68619\,07204\,44341\,56650\,62312\,57307\,27145\,23371\,85859\,25169\,53304\,55204\,03164\,34171\,68712\,60201\,72171\,72825\,93982\,17597\,84770\,20199\,57286\,95047\,49610\,40780\,04870\,02090\,32684\,74828\,74211\,21504\,22562\,54506\,87128\,56837\,13569\,95235\,62847\,28296\,00130\,50103\,42072\,35344\,28402\,01207\,89837\,77823\,60731\,81136\,68753\,84354\,94239\,76305\,58814\,38020\,86993\,25390\,65124\,41050\,53034\,49906\,95514\,34511\,90839\,77979\,10008\,09300\,12129\,04329\,37146\,61240\,80514\,25607\,88507\,98122\,14852\,95384\,61005\,33365\,52966\,88636\,37769\,24886\,56701\,33871\,09186\,11213\,66257\,28522\,34189\,95744\,36261\,24608\,81383\,89040\,39103\,47556\,92955\,05398\,27066\,76308\,91155\,64368\,49477\,96282\,48213\,18982\,19724\,58728\,59148\,86874\,37150\,50913\,87331\,43934\,11326\,43378\,90590\,77061\,58998\,17768\,33014\,23090\,31857\,88496\,81830\,59127\,20085\,51962\,01516\,42029\,15007\,40823\,01080\,00533\,00309\,39446\,77178\,75440\,50622\,39152\,50225\,64804\,04709\,60629\,68846\,85359\,74408\,99934\,22869\,18821\,32079\,87556\,13196\,27792\,09151\,35200\,23003\,06799\,53130\,48886\,93014\,99903\,49526\,60063\,68565\,86012\,52445\,89661\,01252\,16587\,03414\,89802\,15349\,34053\,72500\,92978\,23494\,18564\,07359\,21744\,38229\,90375\,20681\,77138\,28176\,71004\,20574\,91049\,95053\,47137\,26133\,73757\,83167\,26475\,52891\,48022\,05071\,48746\,03377\,20422\,05470\,59003\,99501\,34136\,57320\,01149\,74901\,91705\,72280\,50672\,96291\,16623\,25841\,50517\,61038\,50715\,17465\,63807\,70271\,36701\,51492\,12059\,64151\,57841\,17228\,50703\,72096\,65769\,96944\,21790\,94434\,58628\,98694\,55420\,52788\,17213\,53742\,66993\,17363\,68990\,03818\,05354\,01160\,66617\,43530\,46462\,70416\,56958\,74738\,48354\,62814\,59251\,54486\,23983\,51932\,98400\,38644\,89009\,96908\,19530\,39220\,53429\,66500\,15373\,25249\,40861\,59997\,24080\,87961\,75141\,58253\,00934\,81846\,29286\,48728\,66034\,46820\,68093\,01185\,42220\,33016\,72361\,27557\,97781\,82285\,86757\,27242\,39316\,89786\,88276\,01904\,50817\,89416\,21178\,33401\,93288\,05230\,82487\,59224\,95028\,23139\,57113\,73366\,01266\,23158\,18074\,22154\,69139\,23909\,91505\,29407\,43253\,82597\,40573\,22201\,71483\,13598\,09131\,30381\,95870\,39205\,63395\,10850\,88058\,49011\,69265\,59161\,93555\,44122\,41090\,60053\,25004\,67993\,10239\,84146\,37463\,79029\,83587\,92006\,19505\,88708\,93770\,08735\,26543\,90124\,88785\,03126\,45257\,34217\,70426\,04927\,76552\,83383\,52771\,24892\,65936\,09383\,97637\,50798\,48922\,24035\,84655\,15434\,01428\,58341\,16758\,25501\,03554\,60901\,75655\,02433\,51670\,66365\,14615\,27156\,72111\,22306\,12440\,86896\,74738\,18565\,84631\,20175\,66515\,16753\,34083\,02696\,92173\,74013\,41727\,21587\,49267\,71276\,81241\,86793\,58458\,68236\,26026\,60001\,11617\,67151\,46171\,17344\,06182\,10950\,10285\,25480\,23708\,76594\,84278\,18985\,13148\,10362\,30322\,53898\,70805\,56557\,35180\,70652\,03104\,74230\,51098\,56838\,22945\,71577\,58594\,20389\,50586\,91510\,41904\,71845\,22532\,14233\,22379\,71245\,12543\,09735\,28153\,87062\,65259\,48582\,29607\,84917\,26627\,79027\,94961\,53801\,12002\,18829\,08295\,05218\,83314\,24449\,31369\,32268\,14353\,17342\,24264\,33438\,95563\,45324\,80757\,16186\,11150\,11047\,74404\,17242\,15497\,32231\,99825\,41754\,23415\,60084\,28899\,60784\,67878\,02354\,39265\,43299\,67488\,44279\,05227\,73325\,79052\,07486\,33817\,88831\,04518\,63699\,35213\,81997\,73563\,80230\,58403\,03018\,58735\,97302\,58317\,81150\,81414\,36924\,70549\,89104\,03291\,09530\,27328\,59576\,97403\,57672\,66189\,51041\,24886\,51551\,19119\,02994\,99858\,53614\,69145\,78153\,88613\,87131\,05135\,39191\,41747\,85937\,34983\,25497\,32369\,89862\,12084\,00725\,12206\,55957\,66548\,86372\,83220\,76982\,75789\,56905\,33463\,23927\,55559\,90909\,40142\,06007\,19007\,32569\,84026\,32702\,87110\,48930\,62730\,11638\,28384\,35207\,35557\,60248\,27023\,79019\,34425\,88028\,24866\,23395\,42326\,07958\,80814\,55498\,60205\,87350\,88230\,01638\,29703\,93886\,45916\,52392\,88565\,84635\,84975\,84682\,02701\,32247\,37979\,27035\,65787\,80073\,08286\,76842\,07199\,40647\,60112\,65343\,05710\,31458\,08664\,85760\,52919\,18589\,34650\,33678\,19801\,71890\,56241\,93902\,05938\,92677\,03082\,55477\,97285\,34941\,61370\,30689\,12446\,51495\,47472\,49099\,00019\,97660\,28835\,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Daniel.

P.S.Use a 10-digit tool.As you saw "n=10000" gave 4 decimals.
 
damn it.. i tried quite a few combinations for n upto 350000.49999 but to no avail..

what 10 digit tool are you talking about?

thanks
 
A pocket calculator,u know that small thing which has scientific functions and approximates everything to 10 digits...I don't have one with me here,but i remember that it was useful with these calculations...

Daniel.
 
dextercioby said:
Plug 10000.Here's what Maple had to say...2\,71814\,59268\,25224\,86403\,76646\,74913\,14653\,61138\,22649\,22072\,08183\,70865\,87378\,74197\,73771\,51396\,84730\,52781\,47783\,95201\,39855\,02523\,32514\,27278\,51252\,60669\,48976\,10003\,45427\,31187\,99109\,30676\,84981\,85665\,58206\,31434\,30483\,15259\,13526\,35090\,09396\,74095\,16117\,83827\,54701\,95160\,84152\,56234\,06623\,52454\,30366\,88228\,24462\,79113\,55300\,03628\,76646\,77326\,89431\,89206\,81691\,97525\,60437\,44118\,02619\,08090\,91242\,59954\,30139\,97331\,08900\,22516\,98440\,41752\,78311\,74265\,78254\,09053\,90869\,80995\,48471\,29021\,21665\,66588\,14625\,02271\,09169\,30414\,86593\,95043\,47113\,79360\,52635\,68619\,07204\,44341\,56650\,62312\,57307\,27145\,23371\,85859\,25169\,53304\,55204\,03164\,34171\,68712\,60201\,72171\,72825\,93982\,17597\,84770\,20199\,57286\,95047\,49610\,40780\,04870\,02090\,32684\,74828\,74211\,21504\,22562\,54506\,87128\,56837\,13569\,95235\,62847\,28296\,00130\,50103\,42072\,35344\,28402\,01207\,89837\,77823\,60731\,81136\,68753\,84354\,94239\,76305\,58814\,38020\,86993\,25390\,65124\,41050\,53034\,49906\,95514\,34511\,90839\,77979\,10008\,09300\,12129\,04329\,37146\,61240\,80514\,25607\,88507\,98122\,14852\,95384\,61005\,33365\,52966\,88636\,37769\,24886\,56701\,33871\,09186\,11213\,66257\,28522\,34189\,95744\,36261\,24608\,81383\,89040\,39103\,47556\,92955\,05398\,27066\,76308\,91155\,64368\,49477\,96282\,48213\,18982\,19724\,58728\,59148\,86874\,37150\,50913\,87331\,43934\,11326\,43378\,90590\,77061\,58998\,17768\,33014\,23090\,31857\,88496\,81830\,59127\,20085\,51962\,01516\,42029\,15007\,40823\,01080\,00533\,00309\,39446\,77178\,75440\,50622\,39152\,50225\,64804\,04709\,60629\,68846\,85359\,74408\,99934\,22869\,18821\,32079\,87556\,13196\,27792\,09151\,35200\,23003\,06799\,53130\,48886\,93014\,99903\,49526\,60063\,68565\,86012\,52445\,89661\,01252\,16587\,03414\,89802\,15349\,34053\,72500\,92978\,23494\,18564\,07359\,21744\,38229\,90375\,20681\,77138\,28176\,71004\,20574\,91049\,95053\,47137\,26133\,73757\,83167\,26475\,52891\,48022\,05071\,48746\,03377\,20422\,05470\,59003\,99501\,34136\,57320\,01149\,74901\,91705\,72280\,50672\,96291\,16623\,25841\,50517\,61038\,50715\,17465\,63807\,70271\,36701\,51492\,12059\,64151\,57841\,17228\,50703\,72096\,65769\,96944\,21790\,94434\,58628\,98694\,55420\,52788\,17213\,53742\,66993\,17363\,68990\,03818\,05354\,01160\,66617\,43530\,46462\,70416\,56958\,74738\,48354\,62814\,59251\,54486\,23983\,51932\,98400\,38644\,89009\,96908\,19530\,39220\,53429\,66500\,15373\,25249\,40861\,59997\,24080\,87961\,75141\,58253\,00934\,81846\,29286\,48728\,66034\,46820\,68093\,01185\,42220\,33016\,72361\,27557\,97781\,82285\,86757\,27242\,39316\,89786\,88276\,01904\,50817\,89416\,21178\,33401\,93288\,05230\,82487\,59224\,95028\,23139\,57113\,73366\,01266\,23158\,18074\,22154\,69139\,23909\,91505\,29407\,43253\,82597\,40573\,22201\,71483\,13598\,09131\,30381\,95870\,39205\,63395\,10850\,88058\,49011\,69265\,59161\,93555\,44122\,41090\,60053\,25004\,67993\,10239\,84146\,37463\,79029\,83587\,92006\,19505\,88708\,93770\,08735\,26543\,90124\,88785\,03126\,45257\,34217\,70426\,04927\,76552\,83383\,52771\,24892\,65936\,09383\,97637\,50798\,48922\,24035\,84655\,15434\,01428\,58341\,16758\,25501\,03554\,60901\,75655\,02433\,51670\,66365\,14615\,27156\,72111\,22306\,12440\,86896\,74738\,18565\,84631\,20175\,66515\,16753\,34083\,02696\,92173\,74013\,41727\,21587\,49267\,71276\,81241\,86793\,58458\,68236\,26026\,60001\,11617\,67151\,46171\,17344\,06182\,10950\,10285\,25480\,23708\,76594\,84278\,18985\,13148\,10362\,30322\,53898\,70805\,56557\,35180\,70652\,03104\,74230\,51098\,56838\,22945\,71577\,58594\,20389\,50586\,91510\,41904\,71845\,22532\,14233\,22379\,71245\,12543\,09735\,28153\,87062\,65259\,48582\,29607\,84917\,26627\,79027\,94961\,53801\,12002\,18829\,08295\,05218\,83314\,24449\,31369\,32268\,14353\,17342\,24264\,33438\,95563\,45324\,80757\,16186\,11150\,11047\,74404\,17242\,15497\,32231\,99825\,41754\,23415\,60084\,28899\,60784\,67878\,02354\,39265\,43299\,67488\,44279\,05227\,73325\,79052\,07486\,33817\,88831\,04518\,63699\,35213\,81997\,73563\,80230\,58403\,03018\,58735\,97302\,58317\,81150\,81414\,36924\,70549\,89104\,03291\,09530\,27328\,59576\,97403\,57672\,66189\,51041\,24886\,51551\,19119\,02994\,99858\,53614\,69145\,78153\,88613\,87131\,05135\,39191\,41747\,85937\,34983\,25497\,32369\,89862\,12084\,00725\,12206\,55957\,66548\,86372\,83220\,76982\,75789\,56905\,33463\,23927\,55559\,90909\,40142\,06007\,19007\,32569\,84026\,32702\,87110\,48930\,62730\,11638\,28384\,35207\,35557\,60248\,27023\,79019\,34425\,88028\,24866\,23395\,42326\,07958\,80814\,55498\,60205\,87350\,88230\,01638\,29703\,93886\,45916\,52392\,88565\,84635\,84975\,84682\,02701\,32247\,37979\,27035\,65787\,80073\,08286\,76842\,07199\,40647\,60112\,65343\,05710\,31458\,08664\,85760\,52919\,18589\,34650\,33678\,19801\,71890\,56241\,93902\,05938\,92677\,03082\,55477\,97285\,34941\,61370\,30689\,12446\,51495\,47472\,49099\,00019\,97660\,28835\,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843\,41680\,78348\,52078\,51762\,11357\,00131\,94713\,35220\,24707\,00706\,28974\,83750\,18314\,87412\,86844\,40723\,36317\,57279\,82546\,32211\,25564\,09234\,29646\,22683\,59679\,03323\,28699\,53912\,10829\,26830\,48760\,68111\,21683\,19258\,99218\,70349\,66754\,81360\,40820\,22897\,86471\,02002\,12068\,39205\,73782\,19295\,87774\,51918\,99048\,17485\,28930\,39585\,12562\,30478\,62921\,98207\,72008\,67508\,81479\,25453\,73446\,27126\,32661\,97991\,68830\,80202\,26900\,76430\,53598\,94928\,77457\,07808\,57513\,12719\,24911\,72163\,62243\,27283\,26674\,43750\,11917\,54631\,94899\,14640\,45209\,58978\,24230\,59335\,48737\,71864\,37474\,44933\,32645\,58433\,93005\,38420\,57136\,16862\,89018\,81741\,03656\,17324\,22677\,32900\,92100\,12302\,25975\,24070\,41343\,76388\,26140\,60099\,97828\,02207\,98709\,85481\,65552\,66709\,53566\,60960\,31356\,80732\,28375\,02280\,64868\,41153\,77770\,53712\,01741\,25186\,20598\,48032\,62665\,98141\,77324\,27003\,77846\,08216\,19438\,75510\,90089\,89757\,66580\,01414\,04325\,96682\,33639\,20229\,88424\,09606\,94180\,33137\,88219\,30637\,67795\,24131\,16794\,78910\,67252\,18603\,55019\,07307\,02220\,17290\,29565\,48135\,65150\,74337\,07616\,46466\,93752\,48512\,10403\,89657\,53336\,27313\,61097\,71043\,92669\,09136\,38570\,23660\,11942\,14163\,85559\,55870\,18646\,21859\,88430\,02668\,86350\,06505\,50118\,05130\,84814\,19386\,13667\,09805\,64974\,20083\,89414\,39247\,61146\,21741\,98810\,08454\,63772\,20843\,25959\,24621\,40058\,30298\,81325\,02845\,77349\,63268\,47586\,75928\,13779\,23008\,43617\,00591\,50587\,69829\,26610\,45677\,83529\,67797\,52431\,07008\,93893\,42399\,22132\,56280\,24980\,37265\,01189\,70264\,16280\,44665\,12484\,20622\,69625\,24988\,39363\,43619\,87444\,65131\,17120\,93027\,30622\,68713\,92765\,78980\,87941\,11361\,43043\,82663\,75769\,77188\,05075\,56068\,66796\,90775\,99954\,64583\,25632\,66871\,22618\,58895\,92337\,78606\,51663\,62104\,01230\,26571\,65288\,53186\,71902\,26861\,88729\,63042\,15100\,99361\,82496\,56609\,02271\,53036\,81367\,78573\,00183\,32580\,94878\,34474\,88857\,37613\,60131\,09554\,06862\,94794\,33189\,33735\,95243\,75068\,98513\,43005\,63432\,91964\,54031\,70846\,31286\,27283\,02999\,90285\,15562\,82713\,07735\,47201\,93652\,45903\,19819\,26712\,74583\,27371\,40202\,24088\,69951\,26352\,31789\,41987\,83076\,51637\,89142\,36861\,25837\,47906\,61096\,94919\,89928\,81171\,16440\,72381\,41540\,57236\,08644\,91926\,32939\,51856\,95195\,27054\,19061\,18429\,50635\,94201\,36141\,53301\,90938\,41031\,11168\,44800\,17834\,56652\,09130\,50342\,15452\,75239\,89230\,76646\,93669\,19668\,22593\,16336\,61376\,55879\,49342\,08803\,92192\,21200\,77927\,48195\,04702\,26174\,43237\,57008\,77505\,72879\,57295\,87272\,57194\,19141\,75091\,86078\,78684\,34747\,29300\,99767\,53512\,99342\,58700\,21409\,14927\,22075\,22147\,93985\,36984\,74128\,39389\,59281\,04329\,86392\,18784\,48010\,55473\,60815\,16018\,84215\,87743\,78269\,74364\,15385\,16578\,23662\,73532\,68739\,06286\,48614\,50945\,13093\,01954\,68869\,24680\,14472\,27023\,12574\,92304\,87484\,36959\,98854\,38868\,63042\,53406\,76063\,78939\,41947\,03896\,97479\,69614\,60990\,17014\,75153\,35258\,29465\,46105\,33380\,76875\,75338\,59544\,68021\,55335\,99305\,54351\,90417\,86902\,41151\,19014\,08855\,50536\,08508\,85502\,97350\,28252\,89460\,78246\,76402\,15131\,91085\,71208\,22442\,34918\,97298\,52548\,29302\,84266\,55019\,45040\,23494\,32932\,00846\,13472\,49216\,47509\,78789\,61087\,78427\,50218\,69552\,36070\,73618\,31446\,84220\,03562\,69332\,22080\,64632\,70140\,70101\,24042\,44040\,86485\,78016\,63951\,42323\,63466\,68392\,36371\,87263\,62462\,29516\,74469\,95209\,81254\,08490\,65856\,34495\,34172\,40795\,54663\,78294\,44415\,21731\,12561\,87789\,03250\,59908\,28885\,96618\,26813\,74941\,27864\,71174\,69553\,26500\,62730\,78799\,95522\,88988\,81468\,03357\,74728\,91438\,56891\,53684\,00434\,46465\,95154\,50143\,33877\,19394\,82406\,14943\,18795\,44508\,09209\,71106\,27010\,31552\,77737\,51815\,64094\,45049\,38380\,99916.

Daniel.

P.S.Use a 10-digit tool.As you saw "n=10000" gave 4 decimals.

HOLY CRAP DUDE! :bugeye: :eek:
 
Why did u have to quote...?Wasn't it obvious what u were referring to...?And besides,you could have quoted only a part of it...

Yes,that Maple is quite a guy,huh...?

Daniel.
 
dextercioby said:
Why did u have to quote...?Wasn't it obvious what u were referring to...?And besides,you could have quoted only a part of it...

Yes,that Maple is quite a guy,huh...?

Daniel.

I thought the redundancy of it would be funny I guess
 

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