平方根
平方根，是指自乘結果等於的實數，表示為±（√x），讀作正負根號下x或x的平方根。其中的非負數的平方根稱為算術平方根。正整數的平方根通常是無理數。定義：在分數指數中，依定義，可知開平方運算對乘法滿足分配律，即：注意若n是非負實數且時，因為必定是正數，但有正負兩個解。 應等於±；即(見絕對值)。
平方根若一個數x，它的的平方等於a，即 x²=a,
若x的平方等於a，那么x就叫做a的平方根，即√a￣=x
像加減乘除一樣，求平方根也有自己的豎式運算。以求3的算術平方根為例，過程如右下圖：解得3的算術平方根約為1.732
平方根1、因為每次補數需要補兩位，所以被開方數不只一個數位時，要保證補數不能夾著小數點。例如三位數，必須單獨用百位進行運算，補數時補上十位和個位的數。
2、每一個過渡數都是由上一個過渡數變化而後，上一個過渡數的個位數乘以2，如果需要進位，則往前面進1，然後個位升十位，以此類推，而個位上補上新的運算數字。簡單地講，過渡數27.，是第一次商的1乘以20，把個位上的0用第二次商的7來換，過渡數343是前兩次商的17乘以20=340，其中個位0用第三次商的3來換，第三個過渡數3462是前三次商173乘以20=3460，把個位0用第四次的商2來換，依次類推。
3、誤差值的作用。如果要求精確到更高的小數數位，可以按規則，對誤差值繼續進行運算。
常用平方根
√0 = 0（表示根號0等於0，下同）
√1 = 1
√2 = 1.4142135623731
√3 = 1.73205080756888
√4 = 2
√5 = 2.23606797749979
√6 = 2.44948974278318
√7 = 2.64575131106459
√8 = 2.82842712474619
√9 = 3
√10 = 3.16227766016838
√11 = 3.3166247903554
√12 = 3.46410161513775
√13 = 3.60555127546399
√14 = 3.74165738677394
√15 = 3.87298334620742
√16 = 4
√17 = 4.12310562561766
√18 = 4.24264068711928
√19 = 4.35889894354067
√20 = 4.47213595499958
√21 = 4.58257569495584
√22 = 4.69041575982343
√23 = 4.79583152331272
√24 = 4.89897948556636
√25 = 5
√26 = 5.09901951359278
√27 = 5.19615242270663
√28 = 5.29150262212918
√29 = 5.3851648071345
√30 = 5.47722557505166
√31 = 5.56776436283002
√32 = 5.65685424949238
√33 = 5.74456264653803
√34 = 5.8309518948453
√35 = 5.91607978309962
√36 = 6
√37 = 6.08276253029822
√38 = 6.16441400296898
√39 = 6.2449979983984
√40 = 6.32455532033676
√41 = 6.40312423743285
√42 = 6.48074069840786
√43 = 6.557438524302
√44 = 6.6332495807108
√45 = 6.70820393249937
√46 = 6.78232998312527
√47 = 6.85565460040104
√48 = 6.92820323027551
√49 = 7
√50 = 7.07106781186548
√51 = 7.14142842854285
√52 = 7.21110255092798
√53 = 7.28010988928052
√54 = 7.34846922834953
√55 = 7.41619848709566
√56 = 7.48331477354788
√57 = 7.54983443527075
√58 = 7.61577310586391
√59 = 7.68114574786861
√60 = 7.74596669241483
√61 = 7.81024967590665
√62 = 7.87400787401181
√63 = 7.93725393319377
√64 = 8
√65 = 8.06225774829855
√66 = 8.12403840463596
√67 = 8.18535277187245
√68 = 8.24621125123532
√69 = 8.30662386291807
√70 = 8.36660026534076
√71 = 8.42614977317636
√72 = 8.48528137423857
√73 = 8.54400374531753
√74 = 8.60232526704263
√75 = 8.66025403784439
√76 = 8.71779788708135
√77 = 8.77496438739212
√78 = 8.83176086632785
√79 = 8.88819441731559
√80 = 8.94427190999916
√81 = 9
√82 = 9.05538513813742
√83 = 9.1104335791443
√84 = 9.16515138991168
√85 = 9.21954445729289
√86 = 9.2736184954957
√87 = 9.32737905308882
√88 = 9.38083151964686
√89 = 9.4339811320566
√90 = 9.48683298050514
√91 = 9.53939201416946
√92 = 9.59166304662544
√93 = 9.64365076099295
√94 = 9.69535971483266
√95 = 9.74679434480896
√96 = 9.79795897113271
√97 = 9.8488578017961
√98 = 9.89949493661167
√99 = 9.9498743710662
√100 = 10
√101 = 10.0498756211209
√102 = 10.0995049383621
√103 = 10.1488915650922
√104 = 10.1980390271856
√105 = 10.2469507659596
√106 = 10.295630140987
√107 = 10.3440804327886
√108 = 10.3923048454133
√109 = 10.4403065089106
√110 = 10.4880884817015
√111 = 10.5356537528527
√112 = 10.5830052442584
√113 = 10.6301458127347
√114 = 10.6770782520313
√115 = 10.7238052947636
√116 = 10.770329614269
√117 = 10.816653826392
√118 = 10.8627804912002
√119 = 10.9087121146357
√120 = 10.9544511501033
√121 = 11
√122 = 11.0453610171873
√123 = 11.0905365064094
√124 = 11.13552872566
√125 = 11.1803398874989
√126 = 11.2249721603218
√127 = 11.2694276695846
√128 = 11.3137084989848
√129 = 11.3578166916005
√130 = 11.4017542509914
√131 = 11.4455231422596
√132 = 11.4891252930761
√133 = 11.5325625946708
√134 = 11.5758369027902
√135 = 11.6189500386223
√136 = 11.6619037896906
√137 = 11.7046999107196
√138 = 11.7473401244707
√139 = 11.7898261225516
√140 = 11.8321595661992
√141 = 11.8743420870379
√142 = 11.916375287813
√143 = 11.9582607431014
√144 = 12
√145 = 12.0415945787923
√146 = 12.0830459735946
√147 = 12.1243556529821
√148 = 12.1655250605964
√149 = 12.2065556157337
√150 = 12.2474487139159
√151 = 12.2882057274445
√152 = 12.328828005938
√153 = 12.369316876853
√154 = 12.4096736459909
√155 = 12.4498995979887
√156 = 12.4899959967968
√157 = 12.5299640861417
√158 = 12.5698050899765
√159 = 12.6095202129185
√160 = 12.6491106406735
√161 = 12.6885775404495
√162 = 12.7279220613579
√163 = 12.7671453348037
√164 = 12.8062484748657
√165 = 12.8452325786651
√166 = 12.8840987267251
√167 = 12.9228479833201
√168 = 12.9614813968157
√169 = 13
√170 = 13.0384048104053
√171 = 13.076696830622
√172 = 13.114877048604
√173 = 13.1529464379659
√174 = 13.1909059582729
√175 = 13.228756555323
√176 = 13.2664991614216
√177 = 13.3041346956501
√178 = 13.3416640641263
√179 = 13.3790881602597
√180 = 13.4164078649987
√181 = 13.4536240470737
√182 = 13.490737563232
√183 = 13.5277492584687
√184 = 13.5646599662505
√185 = 13.6014705087354
√186 = 13.6381816969859
√187 = 13.6747943311773
√188 = 13.7113092008021
√189 = 13.7477270848675
√190 = 13.7840487520902
√191 = 13.8202749610853
√192 = 13.856406460551
√193 = 13.8924439894498
√194 = 13.9283882771841
√195 = 13.9642400437689
√196 = 14
√197 = 14.0356688476182
√198 = 14.0712472794703
√199 = 14.1067359796659
√200 = 14.142135623731
√201 = 14.1774468787578
√202 = 14.2126704035519
√203 = 14.247806848775
√204 = 14.2828568570857
√205 = 14.3178210632764
√206 = 14.3527000944073
√207 = 14.3874945699382
√208 = 14.422205101856
√209 = 14.456832294801
√210 = 14.4913767461894
√211 = 14.5258390463339
√212 = 14.560219778561
√213 = 14.5945195193264
√214 = 14.6287388383278
√215 = 14.6628782986152
√216 = 14.6969384566991
√217 = 14.7309198626562
√218 = 14.7648230602334
√219 = 14.7986485869487
√220 = 14.8323969741913
√221 = 14.8660687473185
√222 = 14.8996644257513
√223 = 14.9331845230681
√224 = 14.9666295470958
√225 = 15
√226 = 15.0332963783729
√227 = 15.0665191733194
√228 = 15.0996688705415
√229 = 15.1327459504216
√230 = 15.1657508881031
√231 = 15.1986841535707
√232 = 15.2315462117278
√233 = 15.2643375224737
√234 = 15.2970585407784
√235 = 15.3297097167559
√236 = 15.3622914957372
√237 = 15.3948043183407
√238 = 15.4272486205415
√239 = 15.4596248337403
√240 = 15.4919333848297
√241 = 15.52417469626
√242 = 15.556349186104
√243 = 15.5884572681199
√244 = 15.6204993518133
√245 = 15.6524758424985
√246 = 15.6843871413581
√247 = 15.7162336455017
√248 = 15.7480157480236
√249 = 15.7797338380595
√250 = 15.8113883008419
√251 = 15.8429795177549
√252 = 15.8745078663875
√253 = 15.9059737205869
√254 = 15.9373774505092
√255 = 15.9687194226713
√256 = 16
√257 = 16.0312195418814
√258 = 16.062378404209
√259 = 16.0934769394311
√260 = 16.1245154965971
√261 = 16.1554944214035
√262 = 16.1864140562386
√263 = 16.2172747402269
√264 = 16.2480768092719
√265 = 16.2788205960997
√266 = 16.3095064303001
√267 = 16.3401346383682
√268 = 16.3707055437449
√269 = 16.4012194668567
√270 = 16.431676725155
√271 = 16.4620776331543
√272 = 16.4924225024706
√273 = 16.5227116418583
√274 = 16.5529453572468
√275 = 16.583123951777
√276 = 16.6132477258362
√277 = 16.6433169770932
√278 = 16.6733320005331
√279 = 16.7032930884901
√280 = 16.7332005306815
√281 = 16.7630546142402
√282 = 16.7928556237467
√283 = 16.8226038412607
√284 = 16.8522995463527
√285 = 16.8819430161341
√286 = 16.9115345252878
√287 = 16.9410743460974
√288 = 16.9705627484771
√289 = 17
√290 = 17.0293863659264
√291 = 17.058722109232
√292 = 17.0880074906351
√293 = 17.1172427686237
√294 = 17.1464281994822
√295 = 17.1755640373177
√296 = 17.2046505340853
√297 = 17.2336879396141
√298 = 17.2626765016321
√299 = 17.2916164657906
√300 = 17.3205080756888
√301 = 17.3493515728975
√302 = 17.3781471969828
√303 = 17.4068951855292
√304 = 17.4355957741627
√305 = 17.464249196573
√306 = 17.4928556845359
√307 = 17.5214154679352
√308 = 17.5499287747842
√309 = 17.5783958312469
√310 = 17.606816861659
√311 = 17.6351920885484
√312 = 17.6635217326557
√313 = 17.6918060129541
√314 = 17.7200451466693
√315 = 17.7482393492988
√316 = 17.7763888346312
√317 = 17.8044938147649
√318 = 17.832554500127
√319 = 17.8605710994918
√320 = 17.8885438199983
√321 = 17.9164728671689
√322 = 17.9443584449264
√323 = 17.9722007556114
√324 = 18
√325 = 18.0277563773199
√326 = 18.0554700852678
√327 = 18.0831413200251
√328 = 18.1107702762748
√329 = 18.1383571472171
√330 = 18.1659021245849
√331 = 18.1934053986603
√332 = 18.2208671582886
√333 = 18.2482875908947
√334 = 18.2756668824971
√335 = 18.3030052177231
√336 = 18.3303027798234
√337 = 18.3575597506858
√338 = 18.3847763108502
√339 = 18.411952639522
√340 = 18.4390889145858
√341 = 18.4661853126194
√342 = 18.4932420089069
√343 = 18.5202591774521
√344 = 18.5472369909914
√345 = 18.5741756210067
√346 = 18.6010752377383
√347 = 18.6279360101972
√348 = 18.6547581061776
√349 = 18.6815416922694
√350 = 18.7082869338697
√351 = 18.7349939951952
√352 = 18.7616630392937
√353 = 18.7882942280559
√354 = 18.8148877222268
√355 = 18.8414436814168
√356 = 18.8679622641132
√357 = 18.8944436276912
√358 = 18.9208879284245
√359 = 18.9472953214964
√360 = 18.9736659610103
√361 = 19
√362 = 19.0262975904404
√363 = 19.0525588832577
√364 = 19.0787840283389
√365 = 19.1049731745428
√366 = 19.131126469709
√367 = 19.157244060668
√368 = 19.1833260932509
√369 = 19.2093727122985
√370 = 19.2353840616713
√371 = 19.2613602842582
√372 = 19.2873015219859
√373 = 19.313207915828
√374 = 19.3390796058137
√375 = 19.3649167310371
√376 = 19.3907194296653
√377 = 19.4164878389476
√378 = 19.4422220952236
√379 = 19.4679223339318
√380 = 19.4935886896179
√381 = 19.5192212959431
√382 = 19.5448202856921
√383 = 19.5703857907809
√384 = 19.5959179422654
√385 = 19.6214168703486
√386 = 19.6468827043885
√387 = 19.672315572906
√388 = 19.6977156035922
√389 = 19.723082923316
√390 = 19.7484176581315
√391 = 19.7737199332852
√392 = 19.7989898732233
√393 = 19.824227601599
√394 = 19.8494332412792
√395 = 19.8746069143518
√396 = 19.8997487421324
√397 = 19.9248588451713
√398 = 19.94993734326
√399 = 19.9749843554382
√400 = 20
√401 = 20.0249843945008
√402 = 20.0499376557634
√403 = 20.0748598998847
√404 = 20.0997512422418
√405 = 20.1246117974981
√406 = 20.1494416796099
√407 = 20.174241001832
√408 = 20.1990098767242
√409 = 20.2237484161567
√410 = 20.2484567313166
√411 = 20.2731349327133
√412 = 20.2977831301844
√413 = 20.3224014329016
√414 = 20.3469899493758
√415 = 20.3715487874634
√416 = 20.3960780543711
√417 = 20.4205778566621
√418 = 20.4450483002609
√419 = 20.4694894904587
√420 = 20.4939015319192
√421 = 20.5182845286832
√422 = 20.5426385841741
√423 = 20.5669638012031
√424 = 20.591260281974
√425 = 20.6155281280883
√426 = 20.6397674405503
√427 = 20.6639783197718
√428 = 20.6881608655772
√429 = 20.712315177208
√430 = 20.7364413533277
√431 = 20.7605394920267
√432 = 20.7846096908265
√433 = 20.8086520466848
√434 = 20.8326666559997
√435 = 20.8566536146142
√436 = 20.8806130178211
√437 = 20.9045449603669
√438 = 20.9284495364563
√439 = 20.952326839757
√440 = 20.976176963403
√441 = 21
√442 = 21.0237960416286
√443 = 21.0475651798492
√444 = 21.0713075057055
√445 = 21.095023109729
√446 = 21.1187120819429
√447 = 21.142374511866
√448 = 21.1660104885167
√449 = 21.1896201004171
√450 = 21.2132034355964
√451 = 21.2367605815953
√452 = 21.2602916254693
√453 = 21.2837966537928
√454 = 21.3072757526625
√455 = 21.3307290077015
√456 = 21.3541565040626
√457 = 21.3775583264319
√458 = 21.4009345590327
√459 = 21.4242852856286
√460 = 21.4476105895272
√461 = 21.4709105535839
√462 = 21.4941852602047
√463 = 21.51743479135
√464 = 21.540659228538
√465 = 21.5638586528478
√466 = 21.5870331449229
√467 = 21.6101827849743
√468 = 21.6333076527839
√469 = 21.6564078277077
√470 = 21.6794833886788
√471 = 21.7025344142107
√472 = 21.7255609824004
√473 = 21.7485631709315
√474 = 21.7715410570772
√475 = 21.7944947177034
√476 = 21.8174242292714
√477 = 21.8403296678416
√478 = 21.8632111090754
√479 = 21.8860686282393
√480 = 21.9089023002066
√481 = 21.9317121994613
√482 = 21.9544984001002
√483 = 21.9772609758359
√484 = 22
√485 = 22.0227155455452
√486 = 22.0454076850486
√487 = 22.0680764907139
√488 = 22.0907220343745
√489 = 22.113344387496
√490 = 22.1359436211787
√491 = 22.1585198061603
√492 = 22.1810730128188
√493 = 22.2036033111745
√494 = 22.2261107708929
√495 = 22.248595461287
√496 = 22.2710574513201
√497 = 22.293496809608
√498 = 22.3159136044214
√499 = 22.3383079036887
√500 = 22.3606797749979
