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

[jai6638]

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

 

[dextercioby]

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.

 

[jai6638]

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..

 

[dextercioby]

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.

 

[jai6638]

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

 

[dextercioby]

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.

 

[funkwort]

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!

 

[dextercioby]

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.

 

[funkwort]

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