7546:
5519:
2401:
7246:
2576:
1800:
1790:
45:
6842:
988:
6195:
1341:
4089:
5509:
874:
1218:
3777:
1263:
1103:
3764:
4334:
1491:
1406:
5916:
Starting on the left, bring down the most significant (leftmost) group of digits not yet used (if all the digits have been used, write "0" the number of times required to make a group) and write them to the right of the remainder from the previous step (on the first step, there will be no remainder).
5908:
algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the
5107:
3200:
3579:
983:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
1146:
7452:
3648:
6616:
4173:
1336:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
4084:{\displaystyle {\frac {1}{{\sqrt{a}}+{\sqrt{b}}}}={\frac {{\sqrt{a^{2}}}-{\sqrt{ab}}+{\sqrt{b^{2}}}}{\left({\sqrt{a}}+{\sqrt{b}}\right)\left({\sqrt{a^{2}}}-{\sqrt{ab}}+{\sqrt{b^{2}}}\right)}}={\frac {{\sqrt{a^{2}}}-{\sqrt{ab}}+{\sqrt{b^{2}}}}{a+b}}.}
4670:
1031:
7822:
6966:
257:
1425:
4969:
1348:
7696:
2792:
3027:
3080:
3284:
3489:
1658:
7051:
3639:
3363:
1572:
7130:
7920:
7332:
4855:
7341:
5504:{\displaystyle {\sqrt{z}}={\sqrt{x^{n}+y}}=x+{\cfrac {y}{nx^{n-1}+{\cfrac {(n-1)y}{2x+{\cfrac {(n+1)y}{3nx^{n-1}+{\cfrac {(2n-1)y}{2x+{\cfrac {(2n+1)y}{5nx^{n-1}+{\cfrac {(3n-1)y}{2x+\ddots }}}}}}}}}}}}.}
5879:
4164:
3429:
6510:
1213:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
3085:
2721:
4528:
4439:
1128:
6687:
2019:
1683:
2100:
1244:
8800:
7729:
1516:
3447:
if no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator.
1597:
6883:
3482:
1013:
8519:
7199:
7984:
2501:
Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the
8673:
413:
3768:
When there is a denominator involving surds it is always possible to find a factor to multiply both numerator and denominator by to simplify the expression. For instance using the
516:
181:
8132:
8089:
8047:
2435:
8746:
5661:
4864:
2338:
2284:
1098:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
2535:
8461:
7621:
7221:
5616:
3759:{\displaystyle {\frac {4{\sqrt {2}}}{\sqrt {5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}\cdot {\frac {\sqrt {5}}{\sqrt {5}}}={\frac {4{\sqrt {10}}}{5}}={\frac {4}{5}}{\sqrt {10}}}
8006:
is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that
664:
630:
8871:
6768:
2716:
2612:
2876:
546:
443:
8937:
8900:
8434:
4497:
4468:
2709:
2214:
1926:
759:
579:
378:
8381:
8178:
3217:
6491:
5781:
5564:
4708:
4329:{\displaystyle {\sqrt {3+2{\sqrt {2}}}}={\sqrt {1+2{\sqrt {2}}+2}}={\sqrt {1^{2}+2{\sqrt {2}}+{\sqrt {2}}^{2}}}={\sqrt {\left(1+{\sqrt {2}}\right)^{2}}}=1+{\sqrt {2}}}
2674:
2497:
1876:
8836:
8713:
8632:
8606:
8198:
8004:
7489:
6095:
6046:
2845:
5942:
5700:
8580:
8553:
4370:
1486:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}
6979:
6013:
1401:{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}
7532:
7509:
6175:
6155:
6118:
6069:
5983:
5899:
5740:
5720:
3073:
3053:
2865:
2378:
2358:
2304:
2254:
2234:
3588:
3293:
7064:
7839:
2539:
Since the square of every real number is nonnegative, negative numbers do not have real square roots. However, for every negative real number there are two
7266:
6180:
If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Otherwise go back to step 1 for another iteration.
4758:
1615:
1710:
846:
8148:
th root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of the
1537:
4113:
3195:{\displaystyle {\begin{aligned}{\sqrt{ab}}&={\sqrt{a}}{\sqrt{b}}\\{\sqrt{\frac {a}{b}}}&={\frac {\sqrt{a}}{\sqrt{b}}}\end{aligned}}}
5095:
5571:
3574:{\displaystyle {\sqrt {\frac {32}{5}}}={\sqrt {\frac {16\cdot 2}{5}}}={\sqrt {16}}\cdot {\sqrt {\frac {2}{5}}}=4{\sqrt {\frac {2}{5}}}}
9259:
9067:
6633:
2072:
6334:/ \/ 004 192.000 000 000 (Results) (Explanations) 004 x = 1 10·1·0·
9215:
5786:
9145:
9033:
9004:
6798:
3370:
7223:
to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like
8342:(1824) shows that this is not true in general when the degree is 5 or greater. For example, the solutions of the equation
6801:
to construct a length equal to the square root of a given length, when an auxiliary line of unit length is given. In 1837
3484:
in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:
7447:{\displaystyle \omega =e^{\frac {2\pi i}{n}}=\cos \left({\frac {2\pi }{n}}\right)+i\sin \left({\frac {2\pi }{n}}\right).}
1703:
839:
468:
6253:/ \/ 01 52.27 56 (Results) (Explanations) 01 x = 1 10·1·0·
9367:
9331:
109:
4383:
3431:
strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.
1109:
452:
6230:
4859:
until the desired precision is reached. For computational efficiency, the recurrence relation is commonly rewritten
2510:
1978:
1664:
128:
81:
4097:
can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advanced
6714:
which is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain
1225:
9216:"Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas"
8751:
1497:
5901:. Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows.
1578:
88:
9449:
8347:
6611:{\displaystyle n\log _{b}r=\log _{b}x\quad \quad {\text{hence}}\quad \quad \log _{b}r={\frac {\log _{b}x}{n}}.}
3453:
1696:
994:
832:
798:. Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a
66:
8477:
7178:
6854:
The two square roots of a complex number are always negatives of each other. For example, the square roots of
2643:
2466:
1845:
9408:
9252:
9135:
7939:
8637:
390:
9423:
9418:
6794:
2256:
are integer numerals and the whole expression denotes an irrational number. Irrational numbers of the form
17:
4665:{\displaystyle (1+x)^{\frac {s}{t}}=\sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(s-kt)}{n!t^{n}}}x^{n}}
95:
9444:
9118:
3769:
2382:
31:
8094:
8051:
8009:
4105:
such that the equality of two numbers can be tested by simply looking at their canonical expressions.
2407:
8718:
6974:, then the square root can be obtained by taking the square root of the radius and halving the angle:
5621:
2313:
2259:
6205:
77:
62:
8439:
7817:{\displaystyle {\sqrt{2}},\quad i{\sqrt{2}},\quad -{\sqrt{2}},\quad {\text{and}}\quad -i{\sqrt{2}}.}
7204:
6120:
as the next digit of the root, i.e., above the group of digits you just brought down. Thus the next
5577:
9245:
8339:
805:
639:
605:
8841:
6961:{\displaystyle {\tfrac {1}{\sqrt {2}}}(1+i)\quad {\text{and}}\quad -{\tfrac {1}{\sqrt {2}}}(1+i).}
6717:
2582:
8305:
527:
424:
55:
8913:
8876:
8410:
8200:
is the angle defined in the same way for the number whose root is being taken. Furthermore, all
4473:
4444:
2685:
2190:
1902:
735:
555:
354:
9377:
9341:
8155:
8140:
th roots in the complex plane can be segmented into two steps. First, the magnitude of all the
6350:
y = 1 y = 10·1·0·1 + 10·3·0·1 + 10·3·0·1 = 1 + 0 + 0 =
5909:
radicand. One digit of the root will appear above each group of digits of the original number.
1251:
809:
776:
692:
460:
6460:
5745:
5528:
4677:
9413:
8994:
8808:
8685:
8611:
8585:
8327:
8183:
7989:
7491:. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1,
7463:
6074:
6018:
5513:
2815:
1724:
1222:
825:
5920:
5670:
252:{\displaystyle r^{n}=\underbrace {r\times r\times \dotsb \times r} _{n{\text{ factors}}}=x.}
8558:
8531:
6370:
y = 3096 y = 10·1·1·6 + 10·3·1·6 + 10·3·1·6 = 216 + 1,080 + 1,800 =
5522:
4341:
814:
790:
An unresolved root, especially one using the radical symbol, is sometimes referred to as a
719:
9162:
9075:
6390:
y = 77281 y = 10·1·16·1 + 10·3·16·1 + 10·3·16·1 = 1 + 480 + 76,800 =
8:
8953:
8315:
8281:
5992:
5912:
Beginning with the left-most group of digits, do the following procedure for each group:
4964:{\displaystyle x_{k+1}={\frac {n-1}{n}}\,x_{k}+{\frac {A}{n}}\,{\frac {1}{x_{k}^{n-1}}}.}
4751:
2868:
1748:
780:
775:. The principal root of a positive real number is thus also a positive real number. As a
9049:
7514:
9403:
9282:
8319:
7691:{\displaystyle \eta ,\;\eta \omega ,\;\eta \omega ^{2},\;\ldots ,\;\eta \omega ^{n-1},}
7494:
7143:
6160:
6140:
6103:
6054:
5968:
5884:
5725:
5705:
4738:
3058:
3038:
2850:
2363:
2343:
2289:
2239:
2219:
9196:
6502:
6410:
y = 15571928 y = 10·1·161·2 + 10·3·161·2 + 10·3·161·2 = 8 + 19,320 + 15,552,600 =
102:
9141:
9029:
9022:
9000:
7545:
3035:
th root, and so the rules for operations with surds involving non-negative radicands
2787:{\displaystyle {\begin{aligned}{\sqrt{8}}&=2\\{\sqrt{-8}}&=-2.\end{aligned}}}
2165:
2117:
2064:
1752:
3022:{\displaystyle {\sqrt{a^{m}}}=(a^{m})^{1/n}=a^{m/n}=(a^{1/n})^{m}=({\sqrt{a}})^{m}.}
9322:
9192:
8972:
8389:
8335:
8220:
2540:
2454:
2109:
597:
168:
6212:
5904:
Write the original number in decimal form. The numbers are written similar to the
3279:{\displaystyle {\sqrt {-1}}\times {\sqrt {-1}}\neq {\sqrt {-1\times -1}}=1,\quad }
9387:
4711:
4520:
1892:
1764:
723:
294:
30:
This article is about nth-roots of real and complex numbers. For other uses, see
9297:
9292:
9268:
8948:
8331:
7929:
6822:
6802:
4974:
4102:
4094:
3440:
3209:
2797:
2631:
2553:
2180:
2029:
1941:
1740:
1413:
1134:
772:
680:
673:
456:
381:
164:
3583:
Next, there is a fraction under the radical sign, which we change as follows:
1653:{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
9438:
7240:
7139:
5905:
4098:
1768:
1744:
784:
704:
9183:
Richard, Zippel (1985). "Simplification of
Expressions Involving Radicals".
7046:{\displaystyle {\sqrt {re^{i\theta }}}=\pm {\sqrt {r}}\cdot e^{i\theta /2}.}
9170:
Proceedings of the 1976 ACM Symposium on
Symbolic and Algebraic Computation
9123:(3rd ed.). Cambridge. §1.13 "Quadratic Surds" – §1.14, pp. 19–23.
7932:) of the number whose root is to be taken; if the number can be written as
2124:
1728:
7724:
th roots of unity. For example, the four different fourth roots of 2 are
5514:
Digit-by-digit calculation of principal roots of decimal (base 10) numbers
9372:
9336:
9114:
9068:"radication – Definition of radication in English by Oxford Dictionaries"
7457:
3634:{\displaystyle 4{\sqrt {\frac {2}{5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}}
3358:{\displaystyle \quad {\sqrt {-1}}\times {\sqrt {-1}}=i\times i=i^{2}=-1.}
2395:
1881:
1567:{\displaystyle \scriptstyle {\sqrt{\scriptstyle {\text{radicand}}}}\,=\,}
1019:
283:
142:
6406:≤ 18719000 < 10·1·161·3 + 10·3·161·3 + 10·3·161·3
5518:
9351:
8323:
8311:
8285:
8248:
is another. This is because raising the latter's coefficient –1 to the
7827:
7135:
7125:{\displaystyle {\sqrt {re^{i\theta }}}={\sqrt {r}}\cdot e^{i\theta /2}}
6971:
4101:. Moreover, when complete denesting is impossible, there is no general
2400:
7915:{\displaystyle {\sqrt{re^{i\theta }}}={\sqrt{r}}\cdot e^{i\theta /n}.}
6429:
The desired precision is achieved. The cube root of 4192 is 16.124...
9382:
9307:
9302:
8395:
7549:
Geometric representation of the 2nd to 6th roots of a complex number
7245:
6442:
2575:
2570:
2176:
1743:
and play a fundamental role in various areas of mathematics, such as
1732:
1603:
289:
7327:{\displaystyle 1,\;\omega ,\;\omega ^{2},\;\ldots ,\;\omega ^{n-1},}
7059:
root of a complex number may be chosen in various ways, for example
5066:= 2.02439 74584 99885 04251 08172 45541 93741 91146 21701 07311 8...
44:
9312:
9287:
9211:
9090:
6386:≤ 96000 < 10·1·16·2 + 10·3·16·2 + 10·3·16·2
1799:
1789:
1345:
862:
9237:
691:
th roots, equally distributed around a complex circle of constant
6841:
6625:
4850:{\displaystyle x_{k+1}=x_{k}-{\frac {x_{k}^{n}-A}{nx_{k}^{n-1}}}}
767:
th root with the greatest real part and in the special case when
3643:
Finally, we remove the radical from the denominator as follows:
8326:
could be expressed in terms of a finite number of radicals and
7228:
7224:
6366:≤ 3192 < 10·1·1·7 + 10·3·1·7 + 10·3·1·7
6346:≤ 4 < 10·1·0·2 + 10·3·0·2 + 10·3·0·2
6324:
Find the cube root of 4192 truncated to the nearest thousandth.
1272:
153:
8330:). However, while this is true for third degree polynomials (
6265:
y = 1 y = 10·1·0·1 + 10·2·0·1 = 1 + 0 =
4977:, and to compute once for all the first factor of each term.
1932:
equal to 2 this is called the principal square root and the
6426:≤ 3147072000 < 10·1·1612·5 + 10·3·1612·5 + 10·3·1612·5
6281:
y = 44 y = 10·1·1·2 + 10·2·1·2 = 4 + 40 =
5001:(initial guess). The first 5 iterations are, approximately:
3031:
Every non-negative number has exactly one non-negative real
7175:
Using the first(last) branch cut the principal square root
6297:
y = 729 y = 10·1·12·3 + 10·2·12·3 = 9 + 720 =
1431:
1394:
1269:
1152:
1037:
880:
8280:
As with square roots, the formula above does not define a
6313:
y = 9856 y = 10·1·123·4 + 10·2·123·4 = 16 + 9840 =
5663:, follows a pattern involving Pascal's triangle. For the
3434:
710:, and this circle degenerates to a point.) Extracting the
9091:"Earliest Known Uses of Some of the Words of Mathematics"
5944:
and add the digits from the next group. This will be the
5874:{\displaystyle \sum _{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}}
2543:
square roots. For example, the square roots of −25 are 5
2059:
th powers, and all rationals except the quotients of two
6309:≤ 9856 < 10·1·123·5 + 10·2·123·5
2055:
th roots of almost all numbers (all integers except the
8208:
th roots are at equally spaced angles from each other.
7460:
in the complex plane, at angles which are multiples of
6695:
raised to the power of the result of the division, not
6501:
also positive, one takes logarithms of both sides (any
5881:. For convenience, call the result of this expression
5451:
5421:
5388:
5358:
5341:
5311:
5278:
5251:
5234:
5207:
5177:
5165:
4159:{\displaystyle {\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}.}
3424:{\displaystyle {\sqrt{a}}\times {\sqrt{b}}={\sqrt{ab}}}
8436:
is rational. That is, it can be reduced to a fraction
7208:
7182:
6927:
6888:
5454:
5424:
5391:
5361:
5344:
5314:
5281:
5254:
5237:
5210:
5180:
5168:
4980:
For example, to find the fifth root of 34, we plug in
3457:
2134:), who referred to rational and irrational numbers as
1668:
1619:
1582:
1544:
1541:
1501:
1455:
1437:
1434:
1429:
1383:
1372:
1361:
1358:
1352:
1309:
1302:
1299:
1285:
1278:
1275:
1267:
1229:
1179:
1158:
1155:
1150:
1113:
1064:
1043:
1040:
1035:
998:
949:
928:
907:
886:
883:
878:
8916:
8879:
8844:
8811:
8754:
8721:
8688:
8640:
8614:
8588:
8561:
8534:
8480:
8442:
8413:
8350:
8219:
th roots, of which there are an even number, come in
8186:
8158:
8097:
8054:
8012:
7992:
7942:
7842:
7732:
7624:
7517:
7497:
7466:
7344:
7269:
7207:
7181:
7067:
6982:
6886:
6720:
6636:
6513:
6463:
6163:
6143:
6106:
6077:
6057:
6021:
5995:
5971:
5923:
5887:
5789:
5748:
5728:
5708:
5673:
5624:
5580:
5531:
5110:
4867:
4761:
4680:
4531:
4476:
4447:
4386:
4344:
4176:
4116:
3780:
3651:
3591:
3492:
3456:
3373:
3296:
3220:
3083:
3061:
3041:
2879:
2853:
2847:, makes it easier to manipulate powers and roots. If
2818:
2719:
2688:
2646:
2585:
2513:
2469:
2410:
2366:
2346:
2316:
2292:
2262:
2242:
2222:
2193:
2075:
1981:
1905:
1848:
1667:
1618:
1581:
1540:
1500:
1428:
1351:
1266:
1228:
1149:
1112:
1034:
997:
877:
738:
642:
608:
558:
530:
471:
427:
393:
357:
184:
6293:≤ 827 < 10·1·12·4 + 10·2·12·4
6809:th root of a given length cannot be constructed if
6441:th root of a positive number can be computed using
5989:, ignoring any decimal point. (For the first step,
5094:Newton's method can be modified to produce various
1959:th root, while negative numbers do not have a real
771:is a negative real number, the one with a positive
69:. Unsourced material may be challenged and removed.
9021:
8931:
8894:
8865:
8830:
8794:
8740:
8707:
8667:
8626:
8600:
8574:
8547:
8513:
8455:
8428:
8375:
8192:
8172:
8126:
8083:
8041:
7998:
7978:
7914:
7816:
7690:
7526:
7503:
7483:
7446:
7326:
7215:
7193:
7124:
7045:
6960:
6762:
6681:
6610:
6485:
6169:
6149:
6112:
6089:
6063:
6040:
6007:
5977:
5936:
5893:
5873:
5775:
5734:
5714:
5694:
5655:
5610:
5558:
5503:
4963:
4849:
4702:
4664:
4491:
4462:
4433:
4364:
4328:
4158:
4083:
3758:
3633:
3573:
3476:
3423:
3357:
3278:
3194:
3067:
3047:
3021:
2859:
2839:
2786:
2703:
2668:
2606:
2529:
2491:
2429:
2372:
2352:
2332:
2298:
2278:
2248:
2228:
2208:
2142:, respectively. This later led to the Arabic word
2094:
2013:
1920:
1870:
1677:
1652:
1591:
1566:
1510:
1485:
1400:
1335:
1238:
1212:
1122:
1097:
1007:
982:
753:
658:
624:
573:
540:
510:
437:
407:
372:
251:
8284:over the entire complex plane, but instead has a
6354:003 192 x = 6 10·1·1·
415:. The square root is usually written without the
9436:
6277:≤ 52 < 10·1·1·3 + 10·2·1·3
6261:≤ 1 < 10·1·0·2 + 10·2·0·2
4434:{\displaystyle {\sqrt{r}}={\sqrt{p}}/{\sqrt{q}}}
1123:{\displaystyle \scriptstyle {\text{difference}}}
9028:. Englewood Cliffs, New Jersey: Prentice-Hall.
7928:is the magnitude (the modulus, also called the
6682:{\displaystyle r=b^{{\frac {1}{n}}\log _{b}x}.}
6394:018 719 000 x = 2 10·1·161·
2014:{\displaystyle {\sqrt{-2}}=-1.148698354\ldots }
1975:th root. For example, −2 has a real 5th root,
1678:{\displaystyle \scriptstyle {\text{logarithm}}}
293:. Roots of higher degree are referred by using
6374:096 000 x = 1 10·1·16·
5574:, it can be seen that the formula used there,
4519:The radical or root may be represented by the
2095:{\displaystyle {\sqrt {2}}=1.414213562\ldots }
9253:
6788:
3450:For example, to write the radical expression
3075:are straightforward within the real numbers:
1704:
1239:{\displaystyle \scriptstyle {\text{product}}}
840:
9220:Journal de Mathématiques Pures et Appliquées
8795:{\displaystyle {\frac {a^{n}}{b^{n}}}=a^{n}}
6414:003 147 072 000 x = 4 10·1·1612·
4717:
2803:
2149:
2143:
2048:th roots.) The only complex root of 0 is 0.
1773:An archaic term for the operation of taking
9160:
8385:cannot be expressed in terms of radicals. (
6699:multiplied by the result of the division.)
5572:digit-by-digit calculation of a square root
2159:
1511:{\displaystyle \scriptstyle {\text{power}}}
9260:
9246:
7665:
7658:
7641:
7631:
7616:th roots in the complex plane. These are
7304:
7297:
7283:
7276:
6497:positive and therefore its principal root
6445:. Starting from the equation that defines
6432:
5917:In other words, multiply the remainder by
4710:. This expression can be derived from the
4093:Simplifying radical expressions involving
1784:
1711:
1697:
1592:{\displaystyle \scriptstyle {\text{root}}}
847:
833:
596:is treated as a complex number it has two
592:has no real-valued square roots, but when
27:Arithmetic operation, inverse of nth power
7456:These roots are evenly spaced around the
6269:00 52 x = 2 10·1·1·
6231:Learn how and when to remove this message
4929:
4905:
3477:{\displaystyle \textstyle {\sqrt {32/5}}}
2021:but −2 does not have any real 6th roots.
1648:
1644:
1562:
1558:
1481:
1477:
1331:
1327:
1208:
1204:
1189:
1185:
1168:
1164:
1093:
1089:
1074:
1070:
1053:
1049:
1008:{\displaystyle \scriptstyle {\text{sum}}}
978:
974:
959:
955:
938:
934:
917:
913:
896:
892:
129:Learn how and when to remove this message
9133:
8514:{\displaystyle x={\frac {a^{n}}{b^{n}}}}
7544:
7244:
7194:{\displaystyle \scriptstyle {\sqrt {z}}}
6840:
5517:
4722:
2682:has exactly one real cube root, written
2574:
2399:
1955:, positive numbers also have a negative
1798:
1788:
32:Root (disambiguation) § Mathematics
9209:
9182:
9163:"Simplification of Radical Expressions"
8973:"Lesson Explainer: nth Roots: Integers"
8396:Proof of irrationality for non-perfect
7979:{\displaystyle r={\sqrt {a^{2}+b^{2}}}}
7157:, or along the negative real axis with
6285:08 27 x = 3 10·1·12·
3435:Simplified form of a radical expression
787:, except along the negative real axis.
14:
9437:
9019:
8992:
8668:{\displaystyle {\frac {a^{n}}{b^{n}}}}
8471:are integers without a common factor.
8299:
7261:th roots in the complex plane, namely
6320:Algorithm terminates: Answer is 12.34
2505:, and is denoted with a radical sign:
2044:is real, this count includes any real
1940:th root can also be represented using
408:{\displaystyle {\sqrt {\phantom {x}}}}
9241:
9113:
9013:
7604:. Principal roots are shown in black.
6624:is recovered from this by taking the
6301:98 56 x = 4 10·1·123·
5082:is accurate to 25 decimal places and
4741:, which starts with an initial guess
3770:factorization of the sum of two cubes
3204:Subtleties can occur when taking the
2796:Every real number has two additional
2183:(1551) all used the term to refer to
1824:is a positive integer, is any of the
7834:th root may be found by the formula
6188:
5055:= 2.02439 74584 99885 04251 08172...
4499:are integers, which means that both
4380:coprime and positive integers. Then
4108:For example, it is not obvious that
2812:th root in its exponent form, as in
2557:represents a number whose square is
548:denotes the positive square root of
67:adding citations to reliable sources
38:
9267:
8996:New Approach to CBSE Mathematics IX
8910:th power, this is impossible. Thus
6770:then proceeding as before to find |
5783:, we can rewrite the expression as
5702:is defined as the value of element
2187:, that is, expressions of the form
2144:
1723:Roots are used for determining the
511:{\displaystyle {\sqrt{x}}=x^{1/n}.}
24:
9088:
8999:. Laxmi Publications. p. 25.
6970:If we express a complex number in
4578:
4514:
4168:The above can be derived through:
2310:; irrational numbers of the form
277:of which the root is taken is the
25:
9461:
9233:
9140:. Cengage Learning. p. 470.
8334:) and fourth degree polynomials (
8127:{\displaystyle \tan \theta =b/a.}
8084:{\displaystyle \sin \theta =b/r,}
8042:{\displaystyle \cos \theta =a/r,}
7234:
6836:
6244:Find the square root of 152.2756.
2430:{\displaystyle y=\pm {\sqrt {x}}}
2108:th roots of rational numbers are
9161:Caviness, B. F.; Fateman, R. J.
8741:{\displaystyle {\frac {n}{1}}=n}
6816:
6193:
5656:{\displaystyle x^{2}+20xp\leq c}
4441:is rational if and only if both
2333:{\displaystyle a\pm {\sqrt {b}}}
2279:{\displaystyle \pm {\sqrt {a}},}
2154:, meaning "deaf" or "dumb") for
585:th root. A negative real number
43:
9203:
9185:Journal of Symbolic Computation
9176:
7792:
7786:
7767:
7748:
6922:
6916:
6710:is odd, there is one real root
6560:
6559:
6553:
6552:
5742:of Pascal's Triangle such that
5096:generalized continued fractions
3297:
3275:
2530:{\displaystyle {\sqrt {25}}=5.}
2389:
2158:being translated into Latin as
2123:The term "surd" traces back to
1805:one of which is a negative real
281:A root of degree 2 is called a
54:needs additional citations for
9154:
9127:
9107:
9082:
9060:
9042:
8986:
8965:
8675:is not in simplest form. Thus
8582:must share a common factor if
8456:{\displaystyle {\frac {a}{b}}}
8322:(that is, that all roots of a
7216:{\displaystyle \scriptstyle z}
6952:
6940:
6913:
6901:
6753:
6745:
6731:
6722:
6202:This section needs editing to
5842:
5830:
5764:
5752:
5689:
5677:
5611:{\displaystyle x(20p+x)\leq c}
5599:
5584:
5547:
5535:
5441:
5426:
5378:
5363:
5331:
5316:
5268:
5256:
5224:
5212:
4690:
4682:
4628:
4613:
4545:
4532:
3007:
2991:
2979:
2957:
2916:
2902:
1641:
1633:
305:, etc. The computation of an
13:
1:
9409:Conway chained arrow notation
9197:10.1016/S0747-7171(85)80014-6
9134:McKeague, Charles P. (2011).
8959:
6051:Determine the greatest digit
5987:part of the root found so far
4973:This allows to have only one
3441:non-nested radical expression
2564:
2169:
2128:
726:of this function, called the
714:th roots of a complex number
659:{\displaystyle -i{\sqrt {x}}}
625:{\displaystyle +i{\sqrt {x}}}
9120:A Course of Pure Mathematics
8866:{\displaystyle {\sqrt{x}}=a}
8215:is even, a complex number's
7537:
6799:use compass and straightedge
6795:ancient Greek mathematicians
6763:{\displaystyle |r|^{n}=|x|,}
5070:(All correct digits shown.)
4750:and then iterates using the
2808:Expressing the degree of an
2607:{\displaystyle y={\sqrt{x}}}
2164:(meaning "deaf" or "mute").
1963:th root. For odd values of
7:
8942:
8223:pairs, so that if a number
6184:
4511:th powers of some integer.
2185:unresolved irrational roots
2150:
541:{\displaystyle {\sqrt {x}}}
520:For a positive real number
451:th root of a number is the
438:{\displaystyle {\sqrt {x}}}
10:
9466:
9361:Inverse for right argument
9050:"Definition of RADICATION"
9020:Silver, Howard A. (1986).
8932:{\displaystyle {\sqrt{x}}}
8895:{\displaystyle {\sqrt{x}}}
8429:{\displaystyle {\sqrt{x}}}
8303:
7238:
6872:, and the square roots of
6789:Geometric constructibility
6691:(Note: That formula shows
4492:{\displaystyle {\sqrt{q}}}
4463:{\displaystyle {\sqrt{p}}}
2704:{\displaystyle {\sqrt{x}}}
2568:
2393:
2209:{\displaystyle {\sqrt{r}}}
1921:{\displaystyle {\sqrt{x}}}
1803:The three 3rd roots of −1,
1765:Square root § History
1762:
1758:
754:{\displaystyle {\sqrt{x}}}
718:can thus be taken to be a
581:denotes the positive real
574:{\displaystyle {\sqrt{x}}}
459:, and can be written as a
373:{\displaystyle {\sqrt{x}}}
287:and a root of degree 3, a
29:
9419:Knuth's up-arrow notation
9396:
9360:
9321:
9275:
8376:{\displaystyle x^{5}=x+1}
8173:{\displaystyle \theta /n}
7608:Every complex number has
4718:Computing principal roots
2804:Identities and properties
2380:are rational, are called
2116:th roots of integers are
2036:different complex number
1793:The four 4th roots of −1,
1739:th roots of 1 are called
1610:
1602:
1532:
1521:
1420:
1412:
1258:
1250:
1141:
1133:
1026:
1018:
869:
861:
679:In general, any non-zero
332:is also a square root of
9424:Steinhaus–Moser notation
9024:Algebra and trigonometry
7249:The three 3rd roots of 1
6486:{\displaystyle r^{n}=x,}
6204:comply with Knowledge's
6177:to form a new remainder.
5776:{\displaystyle P(4,1)=4}
5559:{\displaystyle P(4,1)=4}
4703:{\displaystyle |x|<1}
3208:th roots of negative or
2869:non-negative real number
2669:{\displaystyle r^{3}=x.}
2492:{\displaystyle r^{2}=x.}
2306:is rational, are called
2040:th roots. (In the case
1967:, every negative number
1871:{\displaystyle r^{n}=x.}
1828:real or complex numbers
1769:Cube root § History
806:transcendental functions
804:, and if it contains no
779:, the principal root is
687:distinct complex-valued
9054:www.merriam-webster.com
8831:{\displaystyle x=a^{n}}
8708:{\displaystyle 1^{n}=1}
8627:{\displaystyle b\neq 1}
8601:{\displaystyle b\neq 1}
8306:Root-finding algorithms
8193:{\displaystyle \theta }
7999:{\displaystyle \theta }
7484:{\displaystyle 2\pi /n}
6433:Logarithmic calculation
6090:{\displaystyle y\leq c}
6041:{\displaystyle 0^{0}=1}
2840:{\displaystyle x^{1/n}}
1785:Definition and notation
163:(the root) which, when
8933:
8896:
8867:
8832:
8796:
8742:
8709:
8669:
8628:
8602:
8576:
8549:
8515:
8457:
8430:
8377:
8194:
8174:
8128:
8085:
8043:
8000:
7980:
7916:
7818:
7692:
7605:
7528:
7505:
7485:
7448:
7328:
7250:
7217:
7195:
7126:
7047:
6962:
6851:
6764:
6702:For the case in which
6683:
6612:
6487:
6171:
6151:
6114:
6091:
6065:
6042:
6009:
5979:
5938:
5937:{\displaystyle 10^{n}}
5895:
5875:
5816:
5777:
5736:
5716:
5696:
5695:{\displaystyle P(n,i)}
5657:
5612:
5567:
5560:
5505:
5102:th root. For example,
4965:
4851:
4704:
4666:
4612:
4582:
4493:
4464:
4435:
4366:
4330:
4160:
4085:
3760:
3635:
3575:
3478:
3425:
3359:
3280:
3196:
3069:
3049:
3023:
2861:
2841:
2788:
2705:
2670:
2615:
2608:
2531:
2493:
2438:
2431:
2374:
2354:
2334:
2300:
2280:
2250:
2230:
2210:
2160:
2096:
2024:Every non-zero number
2015:
1922:
1887:has a single positive
1872:
1806:
1796:
1795:none of which are real
1679:
1654:
1593:
1568:
1512:
1487:
1402:
1337:
1240:
1214:
1124:
1099:
1009:
984:
810:transcendental numbers
755:
660:
626:
575:
542:
512:
439:
409:
374:
253:
9450:Operations on numbers
9414:Grzegorczyk hierarchy
8993:Bansal, R.K. (2006).
8934:
8902:is an integer. Since
8897:
8868:
8833:
8797:
8743:
8710:
8670:
8629:
8608:. This means that if
8603:
8577:
8575:{\displaystyle b^{n}}
8550:
8548:{\displaystyle a^{n}}
8516:
8458:
8431:
8378:
8328:elementary operations
8256:yields 1: that is, (–
8195:
8175:
8129:
8086:
8044:
8001:
7981:
7917:
7819:
7708:th root, and 1,
7693:
7548:
7529:
7506:
7486:
7449:
7329:
7248:
7218:
7196:
7127:
7048:
6963:
6844:
6813:is not a power of 2.
6765:
6684:
6613:
6503:base of the logarithm
6488:
6172:
6152:
6115:
6092:
6066:
6043:
6010:
5980:
5939:
5896:
5876:
5790:
5778:
5737:
5717:
5697:
5658:
5613:
5561:
5521:
5506:
4966:
4852:
4737:can be computed with
4723:Using Newton's method
4705:
4667:
4586:
4562:
4494:
4465:
4436:
4367:
4365:{\displaystyle r=p/q}
4331:
4161:
4086:
3761:
3636:
3576:
3479:
3426:
3360:
3281:
3197:
3070:
3050:
3024:
2862:
2842:
2789:
2706:
2671:
2609:
2578:
2532:
2503:principal square root
2494:
2432:
2403:
2383:mixed quadratic surds
2375:
2355:
2335:
2301:
2281:
2251:
2231:
2211:
2097:
2016:
1923:
1873:
1802:
1792:
1725:radius of convergence
1680:
1655:
1594:
1569:
1513:
1488:
1403:
1338:
1241:
1215:
1125:
1100:
1010:
985:
826:Arithmetic operations
763:, is taken to be the
756:
661:
627:
576:
543:
513:
440:
410:
375:
254:
9191:(189–210): 189–210.
8914:
8877:
8873:. This implies that
8842:
8809:
8752:
8719:
8686:
8638:
8612:
8586:
8559:
8532:
8478:
8440:
8411:
8348:
8340:Abel–Ruffini theorem
8320:solved algebraically
8316:polynomial equations
8184:
8156:
8095:
8052:
8010:
7990:
7940:
7840:
7730:
7622:
7515:
7495:
7464:
7342:
7267:
7205:
7179:
7065:
6980:
6884:
6845:The square roots of
6718:
6634:
6511:
6461:
6161:
6141:
6104:
6075:
6055:
6019:
5993:
5969:
5921:
5885:
5787:
5746:
5726:
5706:
5671:
5667:th root of a number
5622:
5578:
5529:
5108:
4865:
4759:
4731:th root of a number
4678:
4529:
4474:
4445:
4384:
4342:
4174:
4114:
3778:
3649:
3589:
3490:
3454:
3371:
3294:
3218:
3081:
3059:
3039:
2877:
2851:
2816:
2717:
2686:
2644:
2583:
2511:
2467:
2408:
2364:
2344:
2314:
2308:pure quadratic surds
2290:
2260:
2240:
2220:
2191:
2073:
1979:
1971:has a real negative
1903:
1891:th root, called the
1846:
1665:
1616:
1579:
1538:
1498:
1426:
1349:
1264:
1226:
1147:
1110:
1032:
995:
875:
815:algebraic expression
736:
722:. By convention the
720:multivalued function
640:
606:
556:
528:
469:
425:
401:
391:
355:
320:is a square root of
182:
63:improve this article
9388:Super-logarithm (4)
9347:Root extraction (3)
9072:Oxford Dictionaries
8954:Twelfth root of two
8300:Solving polynomials
8282:continuous function
7146:with the condition
7134:which introduces a
6505:will do) to obtain
6213:improve the content
6008:{\displaystyle p=0}
5453:
5423:
5390:
5360:
5343:
5313:
5280:
5253:
5236:
5209:
5179:
5167:
4955:
4843:
4811:
4752:recurrence relation
1951:For even values of
1899:, which is written
1749:theory of equations
397:
165:raised to the power
9445:Elementary algebra
9404:Ackermann function
9298:Exponentiation (3)
9293:Multiplication (2)
9137:Elementary algebra
8929:
8892:
8863:
8828:
8792:
8738:
8705:
8665:
8624:
8598:
8572:
8545:
8511:
8453:
8426:
8373:
8296:is discontinuous.
8252:th power for even
8190:
8170:
8124:
8081:
8039:
7996:
7976:
7912:
7814:
7688:
7606:
7527:{\displaystyle -i}
7524:
7501:
7481:
7444:
7324:
7251:
7213:
7212:
7191:
7190:
7144:positive real axis
7122:
7043:
6958:
6938:
6899:
6852:
6760:
6679:
6608:
6483:
6167:
6147:
6110:
6087:
6061:
6038:
6005:
5975:
5934:
5891:
5871:
5773:
5732:
5712:
5692:
5653:
5608:
5568:
5556:
5501:
5494:
5489:
5484:
5479:
5474:
5469:
5448:
5385:
5338:
5275:
5231:
5174:
5073:The approximation
4961:
4935:
4847:
4823:
4797:
4700:
4662:
4489:
4460:
4431:
4362:
4326:
4156:
4081:
3756:
3631:
3571:
3474:
3473:
3421:
3355:
3276:
3192:
3190:
3065:
3045:
3019:
2857:
2837:
2784:
2782:
2701:
2678:Every real number
2666:
2616:
2604:
2527:
2489:
2439:
2427:
2370:
2350:
2330:
2296:
2276:
2246:
2226:
2206:
2118:algebraic integers
2092:
2011:
1918:
1868:
1807:
1797:
1675:
1674:
1650:
1649:
1589:
1588:
1564:
1563:
1550:
1508:
1507:
1483:
1482:
1471:
1468:
1450:
1398:
1397:
1392:
1389:
1378:
1367:
1333:
1332:
1321:
1318:
1315:
1308:
1294:
1291:
1284:
1236:
1235:
1210:
1209:
1198:
1195:
1174:
1120:
1119:
1095:
1094:
1083:
1080:
1059:
1005:
1004:
980:
979:
968:
965:
944:
923:
902:
801:radical expression
751:
656:
622:
571:
538:
508:
435:
405:
370:
249:
239:
227:
9432:
9431:
9325:for left argument
9147:978-0-8400-6421-9
9095:Mathematics Pages
9078:on April 3, 2018.
9035:978-0-13-021270-2
9006:978-81-318-0013-3
8927:
8906:is not a perfect
8890:
8855:
8777:
8730:
8663:
8509:
8451:
8424:
7974:
7883:
7868:
7809:
7790:
7781:
7762:
7743:
7504:{\displaystyle i}
7435:
7400:
7371:
7253:The number 1 has
7188:
7152: < 2
7096:
7086:
7014:
7001:
6937:
6936:
6920:
6898:
6897:
6825:other than 0 has
6656:
6603:
6557:
6329:1 6. 1 2 4
6241:
6240:
6233:
6170:{\displaystyle c}
6150:{\displaystyle y}
6113:{\displaystyle x}
6064:{\displaystyle x}
5978:{\displaystyle p}
5894:{\displaystyle y}
5735:{\displaystyle n}
5715:{\displaystyle i}
5523:Pascal's triangle
5496:
5491:
5486:
5481:
5476:
5471:
5452:
5422:
5389:
5359:
5342:
5312:
5279:
5252:
5235:
5208:
5178:
5166:
5151:
5121:
5044:= 2.02439 7458...
4956:
4927:
4903:
4845:
4650:
4556:
4487:
4458:
4429:
4412:
4397:
4324:
4308:
4295:
4271:
4263:
4252:
4224:
4216:
4195:
4193:
4151:
4135:
4133:
4076:
4062:
4040:
4020:
3995:
3987:
3965:
3945:
3916:
3901:
3882:
3860:
3840:
3815:
3812:
3797:
3754:
3747:
3734:
3728:
3712:
3711:
3706:
3695:
3694:
3687:
3671:
3670:
3663:
3629:
3628:
3621:
3605:
3604:
3569:
3568:
3551:
3550:
3536:
3526:
3525:
3503:
3502:
3471:
3443:is said to be in
3419:
3399:
3384:
3319:
3306:
3264:
3242:
3229:
3186:
3185:
3175:
3155:
3149:
3134:
3122:
3103:
3068:{\displaystyle b}
3048:{\displaystyle a}
3004:
2897:
2860:{\displaystyle a}
2765:
2734:
2699:
2602:
2519:
2425:
2373:{\displaystyle b}
2353:{\displaystyle a}
2328:
2299:{\displaystyle a}
2271:
2249:{\displaystyle r}
2229:{\displaystyle n}
2204:
2179:(1202), and then
2166:Gerard of Cremona
2156:irrational number
2110:algebraic numbers
2081:
1997:
1916:
1753:Fourier transform
1721:
1720:
1688:
1687:
1672:
1639:
1627:
1586:
1556:
1554:
1548:
1505:
1465:
1460:
1447:
1442:
1387:
1376:
1365:
1316:
1313:
1306:
1292:
1289:
1282:
1233:
1193:
1183:
1172:
1162:
1117:
1078:
1068:
1057:
1047:
1002:
963:
953:
942:
932:
921:
911:
900:
890:
749:
654:
620:
569:
536:
482:
433:
403:
368:
273:, and the number
236:
200:
198:
139:
138:
131:
113:
16:(Redirected from
9457:
9397:Related articles
9262:
9255:
9248:
9239:
9238:
9228:
9227:
9207:
9201:
9200:
9180:
9174:
9173:
9167:
9158:
9152:
9151:
9131:
9125:
9124:
9111:
9105:
9104:
9102:
9101:
9086:
9080:
9079:
9074:. Archived from
9064:
9058:
9057:
9046:
9040:
9039:
9027:
9017:
9011:
9010:
8990:
8984:
8983:
8981:
8979:
8969:
8938:
8936:
8935:
8930:
8928:
8926:
8918:
8909:
8905:
8901:
8899:
8898:
8893:
8891:
8889:
8881:
8872:
8870:
8869:
8864:
8856:
8854:
8846:
8837:
8835:
8834:
8829:
8827:
8826:
8805:This means that
8801:
8799:
8798:
8793:
8791:
8790:
8778:
8776:
8775:
8766:
8765:
8756:
8747:
8745:
8744:
8739:
8731:
8723:
8714:
8712:
8711:
8706:
8698:
8697:
8679:should equal 1.
8674:
8672:
8671:
8666:
8664:
8662:
8661:
8652:
8651:
8642:
8633:
8631:
8630:
8625:
8607:
8605:
8604:
8599:
8581:
8579:
8578:
8573:
8571:
8570:
8554:
8552:
8551:
8546:
8544:
8543:
8520:
8518:
8517:
8512:
8510:
8508:
8507:
8498:
8497:
8488:
8474:This means that
8470:
8466:
8462:
8460:
8459:
8454:
8452:
8444:
8435:
8433:
8432:
8427:
8425:
8423:
8415:
8390:quintic equation
8382:
8380:
8379:
8374:
8360:
8359:
8288:at points where
8221:additive inverse
8199:
8197:
8196:
8191:
8179:
8177:
8176:
8171:
8166:
8144:th roots is the
8133:
8131:
8130:
8125:
8117:
8090:
8088:
8087:
8082:
8074:
8048:
8046:
8045:
8040:
8032:
8005:
8003:
8002:
7997:
7985:
7983:
7982:
7977:
7975:
7973:
7972:
7960:
7959:
7950:
7921:
7919:
7918:
7913:
7908:
7907:
7903:
7884:
7882:
7874:
7869:
7867:
7862:
7861:
7860:
7844:
7823:
7821:
7820:
7815:
7810:
7808:
7800:
7791:
7788:
7782:
7780:
7772:
7763:
7761:
7753:
7744:
7742:
7734:
7716:, ...
7697:
7695:
7694:
7689:
7684:
7683:
7654:
7653:
7603:
7602:
7598:
7590:
7586:
7585:
7574:
7573:
7561:
7559:
7553:, in polar form
7552:
7533:
7531:
7530:
7525:
7510:
7508:
7507:
7502:
7490:
7488:
7487:
7482:
7477:
7453:
7451:
7450:
7445:
7440:
7436:
7431:
7423:
7405:
7401:
7396:
7388:
7373:
7372:
7367:
7356:
7333:
7331:
7330:
7325:
7320:
7319:
7293:
7292:
7222:
7220:
7219:
7214:
7200:
7198:
7197:
7192:
7189:
7184:
7171:
7170:
7163: <
7162:
7156:
7155:
7131:
7129:
7128:
7123:
7121:
7120:
7116:
7097:
7092:
7087:
7085:
7084:
7069:
7052:
7050:
7049:
7044:
7039:
7038:
7034:
7015:
7010:
7002:
7000:
6999:
6984:
6967:
6965:
6964:
6959:
6939:
6932:
6928:
6921:
6918:
6900:
6893:
6889:
6877:
6871:
6864:
6857:
6784:
6769:
6767:
6766:
6761:
6756:
6748:
6740:
6739:
6734:
6725:
6706:is negative and
6688:
6686:
6685:
6680:
6675:
6674:
6667:
6666:
6657:
6649:
6617:
6615:
6614:
6609:
6604:
6599:
6592:
6591:
6581:
6570:
6569:
6558:
6555:
6545:
6544:
6526:
6525:
6492:
6490:
6489:
6484:
6473:
6472:
6236:
6229:
6225:
6222:
6216:
6197:
6196:
6189:
6176:
6174:
6173:
6168:
6156:
6154:
6153:
6148:
6124:will be the old
6119:
6117:
6116:
6111:
6100:Place the digit
6096:
6094:
6093:
6088:
6070:
6068:
6067:
6062:
6047:
6045:
6044:
6039:
6031:
6030:
6014:
6012:
6011:
6006:
5984:
5982:
5981:
5976:
5943:
5941:
5940:
5935:
5933:
5932:
5900:
5898:
5897:
5892:
5880:
5878:
5877:
5872:
5870:
5869:
5854:
5853:
5826:
5825:
5815:
5804:
5782:
5780:
5779:
5774:
5741:
5739:
5738:
5733:
5721:
5719:
5718:
5713:
5701:
5699:
5698:
5693:
5662:
5660:
5659:
5654:
5634:
5633:
5617:
5615:
5614:
5609:
5570:Building on the
5565:
5563:
5562:
5557:
5510:
5508:
5507:
5502:
5497:
5495:
5493:
5492:
5490:
5488:
5487:
5485:
5483:
5482:
5480:
5478:
5477:
5475:
5473:
5472:
5470:
5468:
5449:
5447:
5419:
5414:
5413:
5386:
5384:
5356:
5339:
5337:
5309:
5304:
5303:
5276:
5274:
5249:
5232:
5230:
5205:
5200:
5199:
5175:
5173:
5163:
5152:
5150:
5145:
5138:
5137:
5127:
5122:
5120:
5112:
5091:is good for 51.
5090:
5081:
5067:
5056:
5045:
5034:
5023:
5012:
5000:
4990:
4970:
4968:
4967:
4962:
4957:
4954:
4943:
4931:
4928:
4920:
4915:
4914:
4904:
4899:
4888:
4883:
4882:
4856:
4854:
4853:
4848:
4846:
4844:
4842:
4831:
4818:
4810:
4805:
4795:
4790:
4789:
4777:
4776:
4749:
4736:
4730:
4709:
4707:
4706:
4701:
4693:
4685:
4671:
4669:
4668:
4663:
4661:
4660:
4651:
4649:
4648:
4647:
4631:
4611:
4600:
4584:
4581:
4576:
4558:
4557:
4549:
4506:
4502:
4498:
4496:
4495:
4490:
4488:
4486:
4478:
4469:
4467:
4466:
4461:
4459:
4457:
4449:
4440:
4438:
4437:
4432:
4430:
4428:
4420:
4418:
4413:
4411:
4403:
4398:
4396:
4388:
4379:
4375:
4371:
4369:
4368:
4363:
4358:
4335:
4333:
4332:
4327:
4325:
4320:
4309:
4307:
4306:
4301:
4297:
4296:
4291:
4277:
4272:
4270:
4269:
4264:
4259:
4253:
4248:
4240:
4239:
4230:
4225:
4217:
4212:
4201:
4196:
4194:
4189:
4178:
4165:
4163:
4162:
4157:
4152:
4147:
4136:
4134:
4129:
4118:
4090:
4088:
4087:
4082:
4077:
4075:
4064:
4063:
4061:
4056:
4055:
4046:
4041:
4039:
4034:
4026:
4021:
4019:
4014:
4013:
4004:
4001:
3996:
3994:
3993:
3989:
3988:
3986:
3981:
3980:
3971:
3966:
3964:
3959:
3951:
3946:
3944:
3939:
3938:
3929:
3922:
3918:
3917:
3915:
3907:
3902:
3900:
3892:
3884:
3883:
3881:
3876:
3875:
3866:
3861:
3859:
3854:
3846:
3841:
3839:
3834:
3833:
3824:
3821:
3816:
3814:
3813:
3811:
3803:
3798:
3796:
3788:
3782:
3765:
3763:
3762:
3757:
3755:
3750:
3748:
3740:
3735:
3730:
3729:
3724:
3718:
3713:
3707:
3702:
3701:
3696:
3690:
3689:
3688:
3683:
3677:
3672:
3666:
3665:
3664:
3659:
3653:
3640:
3638:
3637:
3632:
3630:
3624:
3623:
3622:
3617:
3611:
3606:
3597:
3596:
3580:
3578:
3577:
3572:
3570:
3561:
3560:
3552:
3543:
3542:
3537:
3532:
3527:
3521:
3510:
3509:
3504:
3495:
3494:
3483:
3481:
3480:
3475:
3472:
3467:
3459:
3430:
3428:
3427:
3422:
3420:
3418:
3413:
3405:
3400:
3398:
3390:
3385:
3383:
3375:
3364:
3362:
3361:
3356:
3345:
3344:
3320:
3312:
3307:
3299:
3285:
3283:
3282:
3277:
3265:
3248:
3243:
3235:
3230:
3222:
3212:. For instance:
3201:
3199:
3198:
3193:
3191:
3187:
3184:
3176:
3174:
3166:
3165:
3156:
3154:
3142:
3141:
3135:
3133:
3125:
3123:
3121:
3113:
3104:
3102:
3097:
3089:
3074:
3072:
3071:
3066:
3054:
3052:
3051:
3046:
3028:
3026:
3025:
3020:
3015:
3014:
3005:
3003:
2995:
2987:
2986:
2977:
2976:
2972:
2953:
2952:
2948:
2932:
2931:
2927:
2914:
2913:
2898:
2896:
2891:
2890:
2881:
2866:
2864:
2863:
2858:
2846:
2844:
2843:
2838:
2836:
2835:
2831:
2793:
2791:
2790:
2785:
2783:
2766:
2764:
2759:
2751:
2735:
2733:
2725:
2711:. For example,
2710:
2708:
2707:
2702:
2700:
2698:
2690:
2675:
2673:
2672:
2667:
2656:
2655:
2613:
2611:
2610:
2605:
2603:
2601:
2593:
2560:
2536:
2534:
2533:
2528:
2520:
2515:
2498:
2496:
2495:
2490:
2479:
2478:
2436:
2434:
2433:
2428:
2426:
2421:
2379:
2377:
2376:
2371:
2359:
2357:
2356:
2351:
2339:
2337:
2336:
2331:
2329:
2324:
2305:
2303:
2302:
2297:
2285:
2283:
2282:
2277:
2272:
2267:
2255:
2253:
2252:
2247:
2235:
2233:
2232:
2227:
2215:
2213:
2212:
2207:
2205:
2203:
2195:
2174:
2171:
2163:
2153:
2147:
2146:
2133:
2130:
2101:
2099:
2098:
2093:
2082:
2077:
2067:. For example,
2020:
2018:
2017:
2012:
1998:
1996:
1991:
1983:
1936:is omitted. The
1927:
1925:
1924:
1919:
1917:
1915:
1907:
1877:
1875:
1874:
1869:
1858:
1857:
1813:
1738:
1713:
1706:
1699:
1684:
1682:
1681:
1676:
1673:
1670:
1659:
1657:
1656:
1651:
1640:
1637:
1629:
1628:
1625:
1598:
1596:
1595:
1590:
1587:
1584:
1573:
1571:
1570:
1565:
1557:
1555:
1552:
1549:
1546:
1543:
1517:
1515:
1514:
1509:
1506:
1503:
1492:
1490:
1489:
1484:
1476:
1472:
1467:
1466:
1463:
1461:
1458:
1449:
1448:
1445:
1443:
1440:
1407:
1405:
1404:
1399:
1396:
1393:
1388:
1385:
1377:
1374:
1366:
1363:
1342:
1340:
1339:
1334:
1326:
1322:
1317:
1314:
1311:
1307:
1304:
1301:
1293:
1290:
1287:
1283:
1280:
1277:
1245:
1243:
1242:
1237:
1234:
1231:
1219:
1217:
1216:
1211:
1203:
1199:
1194:
1191:
1184:
1181:
1173:
1170:
1163:
1160:
1129:
1127:
1126:
1121:
1118:
1115:
1104:
1102:
1101:
1096:
1088:
1084:
1079:
1076:
1069:
1066:
1058:
1055:
1048:
1045:
1014:
1012:
1011:
1006:
1003:
1000:
989:
987:
986:
981:
973:
969:
964:
961:
954:
951:
943:
940:
933:
930:
922:
919:
912:
909:
901:
898:
891:
888:
859:
858:
849:
842:
835:
828:
821:
820:
812:it is called an
770:
766:
762:
760:
758:
757:
752:
750:
748:
740:
717:
713:
709:
702:
698:
690:
686:
671:
667:
665:
663:
662:
657:
655:
650:
633:
631:
629:
628:
623:
621:
616:
595:
591:
584:
580:
578:
577:
572:
570:
568:
560:
551:
547:
545:
544:
539:
537:
532:
523:
517:
515:
514:
509:
504:
503:
499:
483:
481:
473:
446:
444:
442:
441:
436:
434:
429:
418:
414:
412:
411:
406:
404:
402:
395:
379:
377:
376:
371:
369:
367:
359:
350:
346:
339:
335:
331:
327:
323:
319:
308:
276:
264:
258:
256:
255:
250:
238:
237:
234:
228:
223:
194:
193:
177:
173:
169:positive integer
162:
158:
149:
134:
127:
123:
120:
114:
112:
71:
47:
39:
21:
9465:
9464:
9460:
9459:
9458:
9456:
9455:
9454:
9435:
9434:
9433:
9428:
9392:
9373:Subtraction (1)
9368:Predecessor (0)
9356:
9337:Subtraction (1)
9332:Predecessor (0)
9317:
9271:
9269:Hyperoperations
9266:
9236:
9231:
9208:
9204:
9181:
9177:
9165:
9159:
9155:
9148:
9132:
9128:
9112:
9108:
9099:
9097:
9087:
9083:
9066:
9065:
9061:
9048:
9047:
9043:
9036:
9018:
9014:
9007:
8991:
8987:
8977:
8975:
8971:
8970:
8966:
8962:
8945:
8939:is irrational.
8922:
8917:
8915:
8912:
8911:
8907:
8903:
8885:
8880:
8878:
8875:
8874:
8850:
8845:
8843:
8840:
8839:
8822:
8818:
8810:
8807:
8806:
8786:
8782:
8771:
8767:
8761:
8757:
8755:
8753:
8750:
8749:
8722:
8720:
8717:
8716:
8693:
8689:
8687:
8684:
8683:
8657:
8653:
8647:
8643:
8641:
8639:
8636:
8635:
8613:
8610:
8609:
8587:
8584:
8583:
8566:
8562:
8560:
8557:
8556:
8539:
8535:
8533:
8530:
8529:
8528:is an integer,
8503:
8499:
8493:
8489:
8487:
8479:
8476:
8475:
8468:
8464:
8443:
8441:
8438:
8437:
8419:
8414:
8412:
8409:
8408:
8405:
8355:
8351:
8349:
8346:
8345:
8308:
8302:
8276:
8269:
8262:
8247:
8240:
8229:
8185:
8182:
8181:
8162:
8157:
8154:
8153:
8113:
8096:
8093:
8092:
8070:
8053:
8050:
8049:
8028:
8011:
8008:
8007:
7991:
7988:
7987:
7968:
7964:
7955:
7951:
7949:
7941:
7938:
7937:
7899:
7892:
7888:
7878:
7873:
7863:
7853:
7849:
7845:
7843:
7841:
7838:
7837:
7804:
7799:
7787:
7776:
7771:
7757:
7752:
7738:
7733:
7731:
7728:
7727:
7673:
7669:
7649:
7645:
7623:
7620:
7619:
7600:
7593:
7592:
7588:
7577:
7576:
7564:
7563:
7555:
7554:
7550:
7543:
7516:
7513:
7512:
7496:
7493:
7492:
7473:
7465:
7462:
7461:
7424:
7422:
7418:
7389:
7387:
7383:
7357:
7355:
7351:
7343:
7340:
7339:
7309:
7305:
7288:
7284:
7268:
7265:
7264:
7243:
7237:
7206:
7203:
7202:
7183:
7180:
7177:
7176:
7168:
7160:
7158:
7153:
7147:
7112:
7105:
7101:
7091:
7077:
7073:
7068:
7066:
7063:
7062:
7030:
7023:
7019:
7009:
6992:
6988:
6983:
6981:
6978:
6977:
6926:
6917:
6887:
6885:
6882:
6881:
6873:
6866:
6859:
6855:
6839:
6819:
6805:proved that an
6791:
6775:
6752:
6744:
6735:
6730:
6729:
6721:
6719:
6716:
6715:
6662:
6658:
6648:
6647:
6643:
6635:
6632:
6631:
6587:
6583:
6582:
6580:
6565:
6561:
6554:
6540:
6536:
6521:
6517:
6512:
6509:
6508:
6468:
6464:
6462:
6459:
6458:
6435:
6427:
6318:
6237:
6226:
6220:
6217:
6210:
6206:Manual of Style
6198:
6194:
6187:
6162:
6159:
6158:
6142:
6139:
6138:
6105:
6102:
6101:
6076:
6073:
6072:
6056:
6053:
6052:
6026:
6022:
6020:
6017:
6016:
5994:
5991:
5990:
5970:
5967:
5966:
5928:
5924:
5922:
5919:
5918:
5886:
5883:
5882:
5859:
5855:
5849:
5845:
5821:
5817:
5805:
5794:
5788:
5785:
5784:
5747:
5744:
5743:
5727:
5724:
5723:
5707:
5704:
5703:
5672:
5669:
5668:
5629:
5625:
5623:
5620:
5619:
5579:
5576:
5575:
5530:
5527:
5526:
5516:
5455:
5450:
5425:
5420:
5418:
5403:
5399:
5392:
5387:
5362:
5357:
5355:
5345:
5340:
5315:
5310:
5308:
5293:
5289:
5282:
5277:
5255:
5250:
5248:
5238:
5233:
5211:
5206:
5204:
5189:
5185:
5181:
5176:
5169:
5164:
5162:
5146:
5133:
5129:
5128:
5126:
5116:
5111:
5109:
5106:
5105:
5089:
5083:
5080:
5074:
5068:
5065:
5059:
5057:
5054:
5048:
5046:
5043:
5037:
5035:
5032:
5026:
5024:
5021:
5015:
5013:
5010:
5004:
4998:
4992:
4981:
4944:
4939:
4930:
4919:
4910:
4906:
4889:
4887:
4872:
4868:
4866:
4863:
4862:
4832:
4827:
4819:
4806:
4801:
4796:
4794:
4785:
4781:
4766:
4762:
4760:
4757:
4756:
4748:
4742:
4739:Newton's method
4732:
4728:
4725:
4720:
4712:binomial series
4689:
4681:
4679:
4676:
4675:
4656:
4652:
4643:
4639:
4632:
4601:
4590:
4585:
4583:
4577:
4566:
4548:
4544:
4530:
4527:
4526:
4521:infinite series
4517:
4515:Infinite series
4504:
4500:
4482:
4477:
4475:
4472:
4471:
4453:
4448:
4446:
4443:
4442:
4424:
4419:
4414:
4407:
4402:
4392:
4387:
4385:
4382:
4381:
4377:
4373:
4354:
4343:
4340:
4339:
4319:
4302:
4290:
4283:
4279:
4278:
4276:
4265:
4258:
4257:
4247:
4235:
4231:
4229:
4211:
4200:
4188:
4177:
4175:
4172:
4171:
4146:
4128:
4117:
4115:
4112:
4111:
4095:nested radicals
4065:
4057:
4051:
4047:
4045:
4035:
4027:
4025:
4015:
4009:
4005:
4003:
4002:
4000:
3982:
3976:
3972:
3970:
3960:
3952:
3950:
3940:
3934:
3930:
3928:
3927:
3923:
3911:
3906:
3896:
3891:
3890:
3886:
3885:
3877:
3871:
3867:
3865:
3855:
3847:
3845:
3835:
3829:
3825:
3823:
3822:
3820:
3807:
3802:
3792:
3787:
3786:
3781:
3779:
3776:
3775:
3749:
3739:
3723:
3719:
3717:
3700:
3682:
3678:
3676:
3658:
3654:
3652:
3650:
3647:
3646:
3616:
3612:
3610:
3595:
3590:
3587:
3586:
3559:
3541:
3531:
3511:
3508:
3493:
3491:
3488:
3487:
3463:
3458:
3455:
3452:
3451:
3445:simplified form
3437:
3414:
3406:
3404:
3394:
3389:
3379:
3374:
3372:
3369:
3368:
3367:Since the rule
3340:
3336:
3311:
3298:
3295:
3292:
3291:
3247:
3234:
3221:
3219:
3216:
3215:
3210:complex numbers
3189:
3188:
3180:
3170:
3164:
3157:
3150:
3140:
3137:
3136:
3129:
3124:
3117:
3112:
3105:
3098:
3090:
3088:
3084:
3082:
3079:
3078:
3060:
3057:
3056:
3040:
3037:
3036:
3010:
3006:
2999:
2994:
2982:
2978:
2968:
2964:
2960:
2944:
2940:
2936:
2923:
2919:
2915:
2909:
2905:
2892:
2886:
2882:
2880:
2878:
2875:
2874:
2852:
2849:
2848:
2827:
2823:
2819:
2817:
2814:
2813:
2806:
2781:
2780:
2767:
2760:
2752:
2750:
2747:
2746:
2736:
2729:
2724:
2720:
2718:
2715:
2714:
2694:
2689:
2687:
2684:
2683:
2651:
2647:
2645:
2642:
2641:
2597:
2592:
2584:
2581:
2580:
2573:
2567:
2558:
2514:
2512:
2509:
2508:
2474:
2470:
2468:
2465:
2464:
2420:
2409:
2406:
2405:
2398:
2392:
2365:
2362:
2361:
2345:
2342:
2341:
2323:
2315:
2312:
2311:
2291:
2288:
2287:
2266:
2261:
2258:
2257:
2241:
2238:
2237:
2221:
2218:
2217:
2199:
2194:
2192:
2189:
2188:
2172:
2131:
2076:
2074:
2071:
2070:
2063:th powers) are
1992:
1984:
1982:
1980:
1977:
1976:
1911:
1906:
1904:
1901:
1900:
1880:Every positive
1853:
1849:
1847:
1844:
1843:
1811:
1804:
1794:
1787:
1771:
1763:Main articles:
1761:
1736:
1717:
1669:
1666:
1663:
1662:
1636:
1624:
1620:
1617:
1614:
1613:
1583:
1580:
1577:
1576:
1551:
1545:
1542:
1539:
1536:
1535:
1502:
1499:
1496:
1495:
1470:
1469:
1462:
1457:
1456:
1452:
1451:
1444:
1439:
1438:
1433:
1430:
1427:
1424:
1423:
1391:
1390:
1384:
1380:
1379:
1373:
1369:
1368:
1362:
1357:
1353:
1350:
1347:
1346:
1320:
1319:
1310:
1303:
1300:
1296:
1295:
1286:
1279:
1276:
1271:
1268:
1265:
1262:
1261:
1230:
1227:
1224:
1223:
1197:
1196:
1190:
1180:
1176:
1175:
1169:
1159:
1154:
1151:
1148:
1145:
1144:
1114:
1111:
1108:
1107:
1082:
1081:
1075:
1065:
1061:
1060:
1054:
1044:
1039:
1036:
1033:
1030:
1029:
999:
996:
993:
992:
967:
966:
960:
950:
946:
945:
939:
929:
925:
924:
918:
908:
904:
903:
897:
887:
882:
879:
876:
873:
872:
853:
824:
768:
764:
744:
739:
737:
734:
733:
731:
724:principal value
715:
711:
707:
700:
696:
688:
684:
669:
649:
641:
638:
637:
635:
615:
607:
604:
603:
601:
593:
586:
582:
564:
559:
557:
554:
553:
549:
531:
529:
526:
525:
521:
495:
491:
487:
477:
472:
470:
467:
466:
428:
426:
423:
422:
420:
416:
396:
394:
392:
389:
388:
363:
358:
356:
353:
352:
348:
344:
337:
333:
329:
325:
321:
317:
311:root extraction
306:
295:ordinal numbers
274:
262:
233:
229:
201:
199:
189:
185:
183:
180:
179:
175:
171:
160:
156:
147:
135:
124:
118:
115:
72:
70:
60:
48:
35:
28:
23:
22:
15:
12:
11:
5:
9463:
9453:
9452:
9447:
9430:
9429:
9427:
9426:
9421:
9416:
9411:
9406:
9400:
9398:
9394:
9393:
9391:
9390:
9385:
9380:
9375:
9370:
9364:
9362:
9358:
9357:
9355:
9354:
9352:Super-root (4)
9349:
9344:
9339:
9334:
9328:
9326:
9319:
9318:
9316:
9315:
9310:
9305:
9300:
9295:
9290:
9285:
9279:
9277:
9273:
9272:
9265:
9264:
9257:
9250:
9242:
9235:
9234:External links
9232:
9230:
9229:
9202:
9175:
9172:. p. 329.
9153:
9146:
9126:
9106:
9089:Miller, Jeff.
9081:
9059:
9041:
9034:
9012:
9005:
8985:
8963:
8961:
8958:
8957:
8956:
8951:
8949:Geometric mean
8944:
8941:
8925:
8921:
8888:
8884:
8862:
8859:
8853:
8849:
8825:
8821:
8817:
8814:
8789:
8785:
8781:
8774:
8770:
8764:
8760:
8737:
8734:
8729:
8726:
8704:
8701:
8696:
8692:
8660:
8656:
8650:
8646:
8623:
8620:
8617:
8597:
8594:
8591:
8569:
8565:
8542:
8538:
8506:
8502:
8496:
8492:
8486:
8483:
8450:
8447:
8422:
8418:
8404:
8394:
8372:
8369:
8366:
8363:
8358:
8354:
8301:
8298:
8274:
8267:
8260:
8245:
8238:
8234:th roots then
8230:is one of the
8227:
8189:
8169:
8165:
8161:
8123:
8120:
8116:
8112:
8109:
8106:
8103:
8100:
8080:
8077:
8073:
8069:
8066:
8063:
8060:
8057:
8038:
8035:
8031:
8027:
8024:
8021:
8018:
8015:
7995:
7971:
7967:
7963:
7958:
7954:
7948:
7945:
7930:absolute value
7911:
7906:
7902:
7898:
7895:
7891:
7887:
7881:
7877:
7872:
7866:
7859:
7856:
7852:
7848:
7813:
7807:
7803:
7798:
7795:
7785:
7779:
7775:
7770:
7766:
7760:
7756:
7751:
7747:
7741:
7737:
7687:
7682:
7679:
7676:
7672:
7668:
7664:
7661:
7657:
7652:
7648:
7644:
7640:
7637:
7634:
7630:
7627:
7542:
7536:
7523:
7520:
7500:
7480:
7476:
7472:
7469:
7443:
7439:
7434:
7430:
7427:
7421:
7417:
7414:
7411:
7408:
7404:
7399:
7395:
7392:
7386:
7382:
7379:
7376:
7370:
7366:
7363:
7360:
7354:
7350:
7347:
7323:
7318:
7315:
7312:
7308:
7303:
7300:
7296:
7291:
7287:
7282:
7279:
7275:
7272:
7239:Main article:
7236:
7235:Roots of unity
7233:
7211:
7187:
7148:0 ≤
7119:
7115:
7111:
7108:
7104:
7100:
7095:
7090:
7083:
7080:
7076:
7072:
7042:
7037:
7033:
7029:
7026:
7022:
7018:
7013:
7008:
7005:
6998:
6995:
6991:
6987:
6957:
6954:
6951:
6948:
6945:
6942:
6935:
6931:
6925:
6915:
6912:
6909:
6906:
6903:
6896:
6892:
6838:
6835:
6823:complex number
6818:
6815:
6803:Pierre Wantzel
6790:
6787:
6759:
6755:
6751:
6747:
6743:
6738:
6733:
6728:
6724:
6678:
6673:
6670:
6665:
6661:
6655:
6652:
6646:
6642:
6639:
6607:
6602:
6598:
6595:
6590:
6586:
6579:
6576:
6573:
6568:
6564:
6551:
6548:
6543:
6539:
6535:
6532:
6529:
6524:
6520:
6516:
6482:
6479:
6476:
6471:
6467:
6437:The principal
6434:
6431:
6327:
6247:
6239:
6238:
6201:
6199:
6192:
6186:
6183:
6182:
6181:
6178:
6166:
6146:
6135:
6134:
6133:
6128:times 10 plus
6109:
6098:
6086:
6083:
6080:
6060:
6049:
6037:
6034:
6029:
6025:
6004:
6001:
5998:
5974:
5962:, as follows:
5952:
5946:current value
5931:
5927:
5890:
5868:
5865:
5862:
5858:
5852:
5848:
5844:
5841:
5838:
5835:
5832:
5829:
5824:
5820:
5814:
5811:
5808:
5803:
5800:
5797:
5793:
5772:
5769:
5766:
5763:
5760:
5757:
5754:
5751:
5731:
5711:
5691:
5688:
5685:
5682:
5679:
5676:
5652:
5649:
5646:
5643:
5640:
5637:
5632:
5628:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5583:
5555:
5552:
5549:
5546:
5543:
5540:
5537:
5534:
5515:
5512:
5500:
5467:
5464:
5461:
5458:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5417:
5412:
5409:
5406:
5402:
5398:
5395:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5354:
5351:
5348:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5307:
5302:
5299:
5296:
5292:
5288:
5285:
5273:
5270:
5267:
5264:
5261:
5258:
5247:
5244:
5241:
5229:
5226:
5223:
5220:
5217:
5214:
5203:
5198:
5195:
5192:
5188:
5184:
5172:
5161:
5158:
5155:
5149:
5144:
5141:
5136:
5132:
5125:
5119:
5115:
5087:
5078:
5063:
5058:
5052:
5047:
5041:
5036:
5033:= 2.02439 7...
5030:
5025:
5019:
5014:
5008:
5003:
4996:
4975:exponentiation
4960:
4953:
4950:
4947:
4942:
4938:
4934:
4926:
4923:
4918:
4913:
4909:
4902:
4898:
4895:
4892:
4886:
4881:
4878:
4875:
4871:
4841:
4838:
4835:
4830:
4826:
4822:
4817:
4814:
4809:
4804:
4800:
4793:
4788:
4784:
4780:
4775:
4772:
4769:
4765:
4746:
4724:
4721:
4719:
4716:
4699:
4696:
4692:
4688:
4684:
4659:
4655:
4646:
4642:
4638:
4635:
4630:
4627:
4624:
4621:
4618:
4615:
4610:
4607:
4604:
4599:
4596:
4593:
4589:
4580:
4575:
4572:
4569:
4565:
4561:
4555:
4552:
4547:
4543:
4540:
4537:
4534:
4516:
4513:
4485:
4481:
4456:
4452:
4427:
4423:
4417:
4410:
4406:
4401:
4395:
4391:
4361:
4357:
4353:
4350:
4347:
4323:
4318:
4315:
4312:
4305:
4300:
4294:
4289:
4286:
4282:
4275:
4268:
4262:
4256:
4251:
4246:
4243:
4238:
4234:
4228:
4223:
4220:
4215:
4210:
4207:
4204:
4199:
4192:
4187:
4184:
4181:
4155:
4150:
4145:
4142:
4139:
4132:
4127:
4124:
4121:
4103:canonical form
4080:
4074:
4071:
4068:
4060:
4054:
4050:
4044:
4038:
4033:
4030:
4024:
4018:
4012:
4008:
3999:
3992:
3985:
3979:
3975:
3969:
3963:
3958:
3955:
3949:
3943:
3937:
3933:
3926:
3921:
3914:
3910:
3905:
3899:
3895:
3889:
3880:
3874:
3870:
3864:
3858:
3853:
3850:
3844:
3838:
3832:
3828:
3819:
3810:
3806:
3801:
3795:
3791:
3785:
3753:
3746:
3743:
3738:
3733:
3727:
3722:
3716:
3710:
3705:
3699:
3693:
3686:
3681:
3675:
3669:
3662:
3657:
3627:
3620:
3615:
3609:
3603:
3600:
3594:
3567:
3564:
3558:
3555:
3549:
3546:
3540:
3535:
3530:
3524:
3520:
3517:
3514:
3507:
3501:
3498:
3470:
3466:
3462:
3436:
3433:
3417:
3412:
3409:
3403:
3397:
3393:
3388:
3382:
3378:
3354:
3351:
3348:
3343:
3339:
3335:
3332:
3329:
3326:
3323:
3318:
3315:
3310:
3305:
3302:
3274:
3271:
3268:
3263:
3260:
3257:
3254:
3251:
3246:
3241:
3238:
3233:
3228:
3225:
3183:
3179:
3173:
3169:
3163:
3160:
3158:
3153:
3148:
3145:
3139:
3138:
3132:
3128:
3120:
3116:
3111:
3108:
3106:
3101:
3096:
3093:
3087:
3086:
3064:
3044:
3018:
3013:
3009:
3002:
2998:
2993:
2990:
2985:
2981:
2975:
2971:
2967:
2963:
2959:
2956:
2951:
2947:
2943:
2939:
2935:
2930:
2926:
2922:
2918:
2912:
2908:
2904:
2901:
2895:
2889:
2885:
2856:
2834:
2830:
2826:
2822:
2805:
2802:
2779:
2776:
2773:
2770:
2768:
2763:
2758:
2755:
2749:
2748:
2745:
2742:
2739:
2737:
2732:
2728:
2723:
2722:
2697:
2693:
2665:
2662:
2659:
2654:
2650:
2600:
2596:
2591:
2588:
2569:Main article:
2566:
2563:
2526:
2523:
2518:
2488:
2485:
2482:
2477:
2473:
2424:
2419:
2416:
2413:
2394:Main article:
2391:
2388:
2369:
2349:
2327:
2322:
2319:
2295:
2275:
2270:
2265:
2245:
2225:
2202:
2198:
2181:Robert Recorde
2091:
2088:
2085:
2080:
2010:
2007:
2004:
2001:
1995:
1990:
1987:
1942:exponentiation
1914:
1910:
1867:
1864:
1861:
1856:
1852:
1786:
1783:
1760:
1757:
1741:roots of unity
1719:
1718:
1716:
1715:
1708:
1701:
1693:
1690:
1689:
1686:
1685:
1660:
1647:
1643:
1638:anti-logarithm
1635:
1632:
1623:
1611:
1608:
1607:
1600:
1599:
1574:
1561:
1533:
1530:
1529:
1519:
1518:
1493:
1480:
1475:
1454:
1453:
1436:
1435:
1432:
1421:
1418:
1417:
1414:Exponentiation
1410:
1409:
1395:
1382:
1381:
1371:
1370:
1360:
1359:
1356:
1343:
1330:
1325:
1298:
1297:
1274:
1273:
1270:
1259:
1256:
1255:
1248:
1247:
1220:
1207:
1202:
1188:
1178:
1177:
1167:
1157:
1156:
1153:
1142:
1139:
1138:
1135:Multiplication
1131:
1130:
1105:
1092:
1087:
1073:
1063:
1062:
1052:
1042:
1041:
1038:
1027:
1024:
1023:
1016:
1015:
990:
977:
972:
958:
948:
947:
937:
927:
926:
916:
906:
905:
895:
885:
884:
881:
870:
867:
866:
855:
854:
852:
851:
844:
837:
829:
773:imaginary part
747:
743:
728:principal root
693:absolute value
681:complex number
674:imaginary unit
653:
648:
645:
619:
614:
611:
600:square roots,
567:
563:
535:
507:
502:
498:
494:
490:
486:
480:
476:
457:exponentiation
432:
400:
382:radical symbol
366:
362:
351:is written as
303:twentieth root
265:is called the
248:
245:
242:
232:
226:
222:
219:
216:
213:
210:
207:
204:
197:
192:
188:
174:, yields
137:
136:
51:
49:
42:
26:
9:
6:
4:
3:
2:
9462:
9451:
9448:
9446:
9443:
9442:
9440:
9425:
9422:
9420:
9417:
9415:
9412:
9410:
9407:
9405:
9402:
9401:
9399:
9395:
9389:
9386:
9384:
9383:Logarithm (3)
9381:
9379:
9376:
9374:
9371:
9369:
9366:
9365:
9363:
9359:
9353:
9350:
9348:
9345:
9343:
9340:
9338:
9335:
9333:
9330:
9329:
9327:
9324:
9320:
9314:
9311:
9309:
9308:Pentation (5)
9306:
9304:
9303:Tetration (4)
9301:
9299:
9296:
9294:
9291:
9289:
9286:
9284:
9283:Successor (0)
9281:
9280:
9278:
9274:
9270:
9263:
9258:
9256:
9251:
9249:
9244:
9243:
9240:
9226:(2): 366–372.
9225:
9221:
9217:
9213:
9206:
9198:
9194:
9190:
9186:
9179:
9171:
9164:
9157:
9149:
9143:
9139:
9138:
9130:
9122:
9121:
9116:
9110:
9096:
9092:
9085:
9077:
9073:
9069:
9063:
9055:
9051:
9045:
9037:
9031:
9026:
9025:
9016:
9008:
9002:
8998:
8997:
8989:
8974:
8968:
8964:
8955:
8952:
8950:
8947:
8946:
8940:
8923:
8919:
8886:
8882:
8860:
8857:
8851:
8847:
8823:
8819:
8815:
8812:
8803:
8787:
8783:
8779:
8772:
8768:
8762:
8758:
8735:
8732:
8727:
8724:
8702:
8699:
8694:
8690:
8680:
8678:
8658:
8654:
8648:
8644:
8621:
8618:
8615:
8595:
8592:
8589:
8567:
8563:
8540:
8536:
8527:
8522:
8504:
8500:
8494:
8490:
8484:
8481:
8472:
8448:
8445:
8420:
8416:
8403:
8399:
8393:
8391:
8388:
8383:
8370:
8367:
8364:
8361:
8356:
8352:
8343:
8341:
8337:
8333:
8329:
8325:
8321:
8317:
8313:
8307:
8297:
8295:
8292: /
8291:
8287:
8283:
8278:
8273:
8266:
8259:
8255:
8251:
8244:
8237:
8233:
8226:
8222:
8218:
8214:
8209:
8207:
8203:
8187:
8167:
8163:
8159:
8151:
8147:
8143:
8139:
8136:Thus finding
8134:
8121:
8118:
8114:
8110:
8107:
8104:
8101:
8098:
8078:
8075:
8071:
8067:
8064:
8061:
8058:
8055:
8036:
8033:
8029:
8025:
8022:
8019:
8016:
8013:
7993:
7969:
7965:
7961:
7956:
7952:
7946:
7943:
7935:
7931:
7927:
7922:
7909:
7904:
7900:
7896:
7893:
7889:
7885:
7879:
7875:
7870:
7864:
7857:
7854:
7850:
7846:
7835:
7833:
7829:
7824:
7811:
7805:
7801:
7796:
7793:
7783:
7777:
7773:
7768:
7764:
7758:
7754:
7749:
7745:
7739:
7735:
7725:
7723:
7719:
7715:
7711:
7707:
7703:
7698:
7685:
7680:
7677:
7674:
7670:
7666:
7662:
7659:
7655:
7650:
7646:
7642:
7638:
7635:
7632:
7628:
7625:
7617:
7615:
7611:
7596:
7584:
7580:
7571:
7567:
7558:
7547:
7540:
7535:
7521:
7518:
7498:
7478:
7474:
7470:
7467:
7459:
7454:
7441:
7437:
7432:
7428:
7425:
7419:
7415:
7412:
7409:
7406:
7402:
7397:
7393:
7390:
7384:
7380:
7377:
7374:
7368:
7364:
7361:
7358:
7352:
7348:
7345:
7337:
7334:
7321:
7316:
7313:
7310:
7306:
7301:
7298:
7294:
7289:
7285:
7280:
7277:
7273:
7270:
7262:
7260:
7256:
7247:
7242:
7241:Root of unity
7232:
7230:
7226:
7209:
7185:
7173:
7167: ≤
7166:
7151:
7145:
7141:
7140:complex plane
7137:
7132:
7117:
7113:
7109:
7106:
7102:
7098:
7093:
7088:
7081:
7078:
7074:
7070:
7060:
7058:
7053:
7040:
7035:
7031:
7027:
7024:
7020:
7016:
7011:
7006:
7003:
6996:
6993:
6989:
6985:
6975:
6973:
6968:
6955:
6949:
6946:
6943:
6933:
6929:
6923:
6910:
6907:
6904:
6894:
6890:
6879:
6876:
6870:
6863:
6850:
6849:
6843:
6834:
6832:
6828:
6824:
6817:Complex roots
6814:
6812:
6808:
6804:
6800:
6796:
6786:
6782:
6778:
6774:|, and using
6773:
6757:
6749:
6741:
6736:
6726:
6713:
6709:
6705:
6700:
6698:
6694:
6689:
6676:
6671:
6668:
6663:
6659:
6653:
6650:
6644:
6640:
6637:
6629:
6627:
6623:
6618:
6605:
6600:
6596:
6593:
6588:
6584:
6577:
6574:
6571:
6566:
6562:
6549:
6546:
6541:
6537:
6533:
6530:
6527:
6522:
6518:
6514:
6506:
6504:
6500:
6496:
6480:
6477:
6474:
6469:
6465:
6456:
6452:
6448:
6444:
6440:
6430:
6425:
6421:
6417:
6413:
6409:
6405:
6401:
6397:
6393:
6389:
6385:
6381:
6377:
6373:
6369:
6365:
6361:
6357:
6353:
6349:
6345:
6341:
6337:
6333:
6330:
6326:
6325:
6321:
6316:
6312:
6308:
6304:
6300:
6296:
6292:
6288:
6284:
6280:
6276:
6272:
6268:
6264:
6260:
6256:
6252:
6250:
6246:
6245:
6235:
6232:
6224:
6214:
6209:
6207:
6200:
6191:
6190:
6179:
6164:
6144:
6136:
6131:
6127:
6123:
6107:
6099:
6084:
6081:
6078:
6058:
6050:
6035:
6032:
6027:
6023:
6002:
5999:
5996:
5988:
5972:
5964:
5963:
5961:
5957:
5953:
5950:
5949:
5929:
5925:
5915:
5914:
5913:
5910:
5907:
5906:long division
5902:
5888:
5866:
5863:
5860:
5856:
5850:
5846:
5839:
5836:
5833:
5827:
5822:
5818:
5812:
5809:
5806:
5801:
5798:
5795:
5791:
5770:
5767:
5761:
5758:
5755:
5749:
5729:
5709:
5686:
5683:
5680:
5674:
5666:
5650:
5647:
5644:
5641:
5638:
5635:
5630:
5626:
5605:
5602:
5596:
5593:
5590:
5587:
5581:
5573:
5553:
5550:
5544:
5541:
5538:
5532:
5524:
5520:
5511:
5498:
5465:
5462:
5459:
5456:
5444:
5438:
5435:
5432:
5429:
5415:
5410:
5407:
5404:
5400:
5396:
5393:
5381:
5375:
5372:
5369:
5366:
5352:
5349:
5346:
5334:
5328:
5325:
5322:
5319:
5305:
5300:
5297:
5294:
5290:
5286:
5283:
5271:
5265:
5262:
5259:
5245:
5242:
5239:
5227:
5221:
5218:
5215:
5201:
5196:
5193:
5190:
5186:
5182:
5170:
5159:
5156:
5153:
5147:
5142:
5139:
5134:
5130:
5123:
5117:
5113:
5103:
5101:
5097:
5092:
5086:
5077:
5071:
5062:
5051:
5040:
5029:
5018:
5007:
5002:
4995:
4988:
4984:
4978:
4976:
4971:
4958:
4951:
4948:
4945:
4940:
4936:
4932:
4924:
4921:
4916:
4911:
4907:
4900:
4896:
4893:
4890:
4884:
4879:
4876:
4873:
4869:
4860:
4857:
4839:
4836:
4833:
4828:
4824:
4820:
4815:
4812:
4807:
4802:
4798:
4791:
4786:
4782:
4778:
4773:
4770:
4767:
4763:
4754:
4753:
4745:
4740:
4735:
4715:
4713:
4697:
4694:
4686:
4672:
4657:
4653:
4644:
4640:
4636:
4633:
4625:
4622:
4619:
4616:
4608:
4605:
4602:
4597:
4594:
4591:
4587:
4573:
4570:
4567:
4563:
4559:
4553:
4550:
4541:
4538:
4535:
4524:
4522:
4512:
4510:
4483:
4479:
4454:
4450:
4425:
4421:
4415:
4408:
4404:
4399:
4393:
4389:
4359:
4355:
4351:
4348:
4345:
4336:
4321:
4316:
4313:
4310:
4303:
4298:
4292:
4287:
4284:
4280:
4273:
4266:
4260:
4254:
4249:
4244:
4241:
4236:
4232:
4226:
4221:
4218:
4213:
4208:
4205:
4202:
4197:
4190:
4185:
4182:
4179:
4169:
4166:
4153:
4148:
4143:
4140:
4137:
4130:
4125:
4122:
4119:
4109:
4106:
4104:
4100:
4099:Galois theory
4096:
4091:
4078:
4072:
4069:
4066:
4058:
4052:
4048:
4042:
4036:
4031:
4028:
4022:
4016:
4010:
4006:
3997:
3990:
3983:
3977:
3973:
3967:
3961:
3956:
3953:
3947:
3941:
3935:
3931:
3924:
3919:
3912:
3908:
3903:
3897:
3893:
3887:
3878:
3872:
3868:
3862:
3856:
3851:
3848:
3842:
3836:
3830:
3826:
3817:
3808:
3804:
3799:
3793:
3789:
3783:
3773:
3771:
3766:
3751:
3744:
3741:
3736:
3731:
3725:
3720:
3714:
3708:
3703:
3697:
3691:
3684:
3679:
3673:
3667:
3660:
3655:
3644:
3641:
3625:
3618:
3613:
3607:
3601:
3598:
3592:
3584:
3581:
3565:
3562:
3556:
3553:
3547:
3544:
3538:
3533:
3528:
3522:
3518:
3515:
3512:
3505:
3499:
3496:
3485:
3468:
3464:
3460:
3448:
3446:
3442:
3432:
3415:
3410:
3407:
3401:
3395:
3391:
3386:
3380:
3376:
3365:
3352:
3349:
3346:
3341:
3337:
3333:
3330:
3327:
3324:
3321:
3316:
3313:
3308:
3303:
3300:
3289:
3288:but, rather,
3286:
3272:
3269:
3266:
3261:
3258:
3255:
3252:
3249:
3244:
3239:
3236:
3231:
3226:
3223:
3213:
3211:
3207:
3202:
3181:
3177:
3171:
3167:
3161:
3159:
3151:
3146:
3143:
3130:
3126:
3118:
3114:
3109:
3107:
3099:
3094:
3091:
3076:
3062:
3042:
3034:
3029:
3016:
3011:
3000:
2996:
2988:
2983:
2973:
2969:
2965:
2961:
2954:
2949:
2945:
2941:
2937:
2933:
2928:
2924:
2920:
2910:
2906:
2899:
2893:
2887:
2883:
2872:
2870:
2854:
2832:
2828:
2824:
2820:
2811:
2801:
2799:
2794:
2777:
2774:
2771:
2769:
2761:
2756:
2753:
2743:
2740:
2738:
2730:
2726:
2712:
2695:
2691:
2681:
2676:
2663:
2660:
2657:
2652:
2648:
2639:
2637:
2633:
2629:
2625:
2621:
2598:
2594:
2589:
2586:
2577:
2572:
2562:
2556:
2555:
2550:
2546:
2542:
2537:
2524:
2521:
2516:
2506:
2504:
2499:
2486:
2483:
2480:
2475:
2471:
2462:
2460:
2456:
2452:
2448:
2444:
2422:
2417:
2414:
2411:
2402:
2397:
2387:
2385:
2384:
2367:
2347:
2325:
2320:
2317:
2309:
2293:
2273:
2268:
2263:
2243:
2223:
2200:
2196:
2186:
2182:
2178:
2167:
2162:
2157:
2152:
2141:
2137:
2126:
2121:
2119:
2115:
2111:
2107:
2102:
2089:
2086:
2083:
2078:
2068:
2066:
2062:
2058:
2054:
2049:
2047:
2043:
2039:
2035:
2031:
2027:
2022:
2008:
2005:
2002:
1999:
1993:
1988:
1985:
1974:
1970:
1966:
1962:
1958:
1954:
1949:
1947:
1943:
1939:
1935:
1931:
1912:
1908:
1898:
1896:
1890:
1886:
1883:
1878:
1865:
1862:
1859:
1854:
1850:
1841:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1801:
1791:
1782:
1780:
1776:
1770:
1766:
1756:
1754:
1750:
1746:
1745:number theory
1742:
1734:
1730:
1726:
1714:
1709:
1707:
1702:
1700:
1695:
1694:
1692:
1691:
1661:
1645:
1630:
1621:
1612:
1609:
1605:
1601:
1575:
1559:
1534:
1531:
1527:
1525:
1520:
1494:
1478:
1473:
1422:
1419:
1415:
1411:
1408:
1354:
1344:
1328:
1323:
1260:
1257:
1253:
1249:
1246:
1221:
1205:
1200:
1186:
1165:
1143:
1140:
1136:
1132:
1106:
1090:
1085:
1071:
1050:
1028:
1025:
1021:
1017:
991:
975:
970:
956:
935:
914:
893:
871:
868:
864:
860:
857:
856:
850:
845:
843:
838:
836:
831:
830:
827:
823:
822:
819:
817:
816:
811:
807:
803:
802:
797:
793:
788:
786:
785:complex plane
783:in the whole
782:
778:
774:
745:
741:
729:
725:
721:
706:
703:is zero with
694:
682:
677:
675:
651:
646:
643:
617:
612:
609:
599:
590:
565:
561:
533:
518:
505:
500:
496:
492:
488:
484:
478:
474:
464:
462:
458:
455:operation of
454:
450:
447:. Taking the
430:
398:
387:
383:
364:
360:
341:
316:For example,
314:
312:
309:th root is a
304:
300:
296:
292:
291:
286:
285:
280:
272:
268:
259:
246:
243:
240:
235: factors
230:
224:
220:
217:
214:
211:
208:
205:
202:
195:
190:
186:
170:
166:
155:
151:
144:
133:
130:
122:
111:
108:
104:
101:
97:
94:
90:
87:
83:
80: –
79:
75:
74:Find sources:
68:
64:
58:
57:
52:This article
50:
46:
41:
40:
37:
33:
19:
9378:Division (2)
9346:
9342:Division (2)
9313:Hexation (6)
9288:Addition (1)
9223:
9219:
9205:
9188:
9184:
9178:
9169:
9156:
9136:
9129:
9119:
9115:Hardy, G. H.
9109:
9098:. Retrieved
9094:
9084:
9076:the original
9071:
9062:
9053:
9044:
9023:
9015:
8995:
8988:
8976:. Retrieved
8967:
8804:
8681:
8676:
8525:
8523:
8473:
8407:Assume that
8406:
8401:
8397:
8386:
8384:
8344:
8310:It was once
8309:
8293:
8289:
8279:
8271:
8264:
8257:
8253:
8249:
8242:
8235:
8231:
8224:
8216:
8212:
8210:
8205:
8201:
8152:th roots is
8149:
8145:
8141:
8137:
8135:
7933:
7925:
7923:
7836:
7831:
7825:
7726:
7721:
7717:
7713:
7709:
7705:
7704:is a single
7701:
7699:
7618:
7613:
7609:
7607:
7594:
7582:
7578:
7569:
7565:
7556:
7538:
7455:
7338:
7335:
7263:
7258:
7254:
7252:
7174:
7164:
7149:
7133:
7061:
7056:
7054:
6976:
6969:
6880:
6874:
6868:
6861:
6853:
6847:
6846:
6837:Square roots
6830:
6826:
6820:
6810:
6806:
6797:knew how to
6792:
6780:
6776:
6771:
6711:
6707:
6703:
6701:
6696:
6692:
6690:
6630:
6621:
6619:
6507:
6498:
6494:
6454:
6450:
6446:
6438:
6436:
6428:
6423:
6422:+ 10·3·1612·
6419:
6418:+ 10·3·1612·
6415:
6411:
6408:015 571 928
6407:
6403:
6399:
6395:
6391:
6387:
6383:
6379:
6375:
6371:
6367:
6363:
6359:
6355:
6351:
6347:
6343:
6339:
6335:
6331:
6328:
6323:
6322:
6319:
6314:
6310:
6306:
6302:
6298:
6294:
6290:
6286:
6282:
6278:
6274:
6270:
6266:
6262:
6258:
6254:
6251:
6248:
6243:
6242:
6227:
6218:
6211:Please help
6203:
6129:
6125:
6121:
5986:
5959:
5955:
5947:
5945:
5911:
5903:
5664:
5569:
5104:
5099:
5093:
5084:
5075:
5072:
5069:
5060:
5049:
5038:
5027:
5016:
5005:
4993:
4986:
4982:
4979:
4972:
4861:
4858:
4755:
4743:
4733:
4726:
4673:
4525:
4518:
4508:
4337:
4170:
4167:
4110:
4107:
4092:
3774:
3767:
3645:
3642:
3585:
3582:
3486:
3449:
3444:
3438:
3366:
3290:
3287:
3214:
3205:
3203:
3077:
3032:
3030:
2873:
2809:
2807:
2800:cube roots.
2795:
2713:
2679:
2677:
2640:
2635:
2627:
2626:is a number
2623:
2622:of a number
2619:
2617:
2552:
2548:
2544:
2538:
2507:
2502:
2500:
2463:
2458:
2453:which, when
2450:
2449:is a number
2446:
2445:of a number
2442:
2440:
2390:Square roots
2381:
2307:
2184:
2155:
2139:
2135:
2125:Al-Khwarizmi
2122:
2113:
2105:
2103:
2069:
2060:
2056:
2052:
2050:
2045:
2041:
2037:
2033:
2025:
2023:
1972:
1968:
1964:
1960:
1956:
1952:
1950:
1945:
1937:
1933:
1929:
1894:
1888:
1884:
1879:
1842:
1837:
1836:th power is
1833:
1829:
1825:
1821:
1817:
1816:of a number
1810:
1808:
1778:
1777:th roots is
1774:
1772:
1729:power series
1722:
1523:
1522:
1192:multiplicand
813:
800:
799:
795:
791:
789:
730:and denoted
727:
705:multiplicity
678:
588:
519:
465:
448:
385:
342:
315:
310:
302:
298:
288:
282:
278:
270:
266:
261:The integer
260:
159:is a number
146:
140:
125:
119:October 2022
116:
106:
99:
92:
85:
73:
61:Please help
56:verification
53:
36:
18:Seventh root
9214:L. (1837).
8312:conjectured
8263:) = (–1) ×
7830:, a single
7458:unit circle
6453:th root of
6402:+ 10·3·161·
6398:+ 10·3·161·
6305:+ 10·2·123·
6249:1 2. 3 4
2443:square root
2396:Square root
2216:, in which
2173: 1150
2087:1.414213562
2006:1.148698354
1882:real number
1312:denominator
1020:Subtraction
699:th root of
347:th root of
299:fourth root
284:square root
143:mathematics
9439:Categories
9100:2008-11-30
8960:References
8838:and thus,
8324:polynomial
8304:See also:
8286:branch cut
7828:polar form
7612:different
7511:, −1, and
7257:different
7142:along the
7136:branch cut
6972:polar form
6833:th roots.
6829:different
6443:logarithms
6412:15,571,928
6382:+ 10·3·16·
6378:+ 10·3·16·
6358:+ 10·3·1·
6338:+ 10·3·0·
6289:+ 10·2·12·
6221:April 2022
6071:such that
2579:The graph
2565:Cube roots
2457:, becomes
2404:The graph
2132: 825
2112:, and all
2065:irrational
2028:, real or
1893:principal
1779:radication
1182:multiplier
1116:difference
1077:subtrahend
781:continuous
463:exponent:
461:fractional
380:using the
89:newspapers
78:"Nth root"
9210:Wantzel,
8619:≠
8593:≠
8400:th power
8318:could be
8314:that all
8188:θ
8160:θ
8105:θ
8102:
8062:θ
8059:
8020:θ
8017:
7994:θ
7897:θ
7886:⋅
7858:θ
7794:−
7769:−
7678:−
7671:ω
7667:η
7660:…
7647:ω
7643:η
7636:ω
7633:η
7626:η
7591:is real,
7519:−
7471:π
7429:π
7416:
7394:π
7381:
7362:π
7346:ω
7314:−
7307:ω
7299:…
7286:ω
7278:ω
7110:θ
7099:⋅
7082:θ
7057:principal
7028:θ
7017:⋅
7007:±
6997:θ
6924:−
6669:
6620:The root
6594:
6572:
6547:
6528:
6457:, namely
6362:+ 10·3·1·
6342:+ 10·3·0·
6273:+ 10·2·1·
6257:+ 10·2·0·
6137:Subtract
6082:≤
5864:−
5810:−
5792:∑
5648:≤
5603:≤
5466:⋱
5436:−
5408:−
5326:−
5298:−
5219:−
5194:−
4949:−
4894:−
4837:−
4813:−
4792:−
4620:−
4606:−
4588:∏
4579:∞
4564:∑
4023:−
3948:−
3843:−
3698:⋅
3539:⋅
3516:⋅
3387:×
3350:−
3328:×
3314:−
3309:×
3301:−
3259:−
3256:×
3250:−
3245:≠
3237:−
3232:×
3224:−
2775:−
2754:−
2620:cube root
2571:Cube root
2541:imaginary
2418:±
2321:±
2264:±
2177:Fibonacci
2140:inaudible
2090:…
2009:…
2003:−
1986:−
1733:root test
1731:with the
1671:logarithm
1631:
1604:Logarithm
1305:numerator
1187:×
1166:×
1072:−
1051:−
644:−
598:imaginary
290:cube root
279:radicand.
225:⏟
218:×
215:⋯
212:×
206:×
9117:(1921).
8943:See also
8463:, where
8336:quartics
8180:, where
7986:. Also,
7720:are the
7572: |
7541:th roots
6388:077 281
6368:003 096
6185:Examples
5525:showing
5098:for the
2551:, where
2340:, where
1820:, where
1547:radicand
1446:exponent
1375:quotient
1364:fraction
1281:dividend
1252:Division
863:Addition
777:function
668:, where
419:as just
338:(−3) = 9
336:, since
324:, since
297:, as in
9323:Inverse
9276:Primary
8978:22 July
8338:), the
8204:of the
7712:,
7560:
7138:in the
6626:antilog
5985:be the
5722:in row
5022:= 2.025
4372:, with
2798:complex
2455:squared
2136:audible
2030:complex
1897:th root
1814:th root
1759:History
1526:th root
1288:divisor
1232:product
1067:minuend
920:summand
910:summand
796:radical
761:
732:
695:. (The
672:is the
666:
636:
632:
602:
453:inverse
445:
421:
167:of the
150:th root
103:scholar
9144:
9032:
9003:
8682:Since
8524:Since
8332:cubics
7700:where
7595:φ
7581:= arg
7579:φ
7562:where
7336:where
7229:Scilab
7225:Matlab
6821:Every
6449:as an
6392:77,281
6317:00 00
6311:98 56
6295:07 29
6279:00 44
2630:whose
2547:and −5
2286:where
2161:surdus
2032:, has
1928:. For
1832:whose
1767:, and
1751:, and
1735:. The
1553:degree
1171:factor
1161:factor
962:addend
952:augend
941:addend
931:addend
328:, and
271:degree
154:number
105:
98:
91:
84:
76:
9166:(PDF)
7936:then
7924:Here
7587:. If
7201:maps
6556:hence
6493:with
6372:3,096
6157:from
5954:Find
5618:, or
4985:= 5,
4674:with
2867:is a
2151:asamm
1727:of a
1606:(log)
1504:power
1464:power
1386:ratio
794:or a
386:radix
326:3 = 9
267:index
152:of a
145:, an
110:JSTOR
96:books
9142:ISBN
9030:ISBN
9001:ISBN
8980:2023
8715:and
8555:and
8467:and
8241:= –
8091:and
7934:a+bi
7575:and
6878:are
6865:and
6858:are
6793:The
6779:= −|
6348:001
6315:9856
6015:and
5965:Let
5958:and
4991:and
4989:= 34
4727:The
4695:<
4507:are
4503:and
4470:and
4376:and
4338:Let
3055:and
2632:cube
2360:and
2236:and
2138:and
2104:All
2051:The
1809:An
1626:base
1585:root
1459:base
1441:base
1056:term
1046:term
899:term
889:term
792:surd
683:has
634:and
552:and
343:The
82:news
9193:doi
8387:cf.
8211:If
8099:tan
8056:sin
8014:cos
7826:In
7789:and
7599:or
7597:= 0
7568:= |
7413:sin
7378:cos
7227:or
6919:and
6660:log
6585:log
6563:log
6538:log
6519:log
6299:729
6263:01
5011:= 2
4999:= 2
2634:is
2175:),
2145:أصم
1944:as
1622:log
1528:(√)
1416:(^)
1254:(÷)
1137:(×)
1022:(−)
1001:sum
865:(+)
808:or
384:or
269:or
141:In
65:by
9441::
9222:.
9218:.
9212:M.
9187:.
9168:.
9093:.
9070:.
9052:.
8802:.
8748:,
8634:,
8521:.
8392:)
8277:.
8270:=
7557:re
7534:.
7231:.
7172:.
7055:A
6867:−2
6856:−4
6785:.
6628::
6283:44
6048:).
5926:10
5819:10
5639:20
5588:20
4714:.
4523::
3772::
3752:10
3726:10
3534:16
3513:16
3497:32
3461:32
3439:A
3353:1.
2871:,
2778:2.
2638::
2618:A
2561:.
2559:−1
2525:5.
2517:25
2461::
2441:A
2386:.
2170:c.
2129:c.
2120:.
1948:.
1840::
1781:.
1755:.
1747:,
818:.
676:.
524:,
340:.
330:−3
313:.
301:,
178::
9261:e
9254:t
9247:v
9224:1
9199:.
9195::
9189:1
9150:.
9103:.
9056:.
9038:.
9009:.
8982:.
8924:n
8920:x
8908:n
8904:x
8887:n
8883:x
8861:a
8858:=
8852:n
8848:x
8824:n
8820:a
8816:=
8813:x
8788:n
8784:a
8780:=
8773:n
8769:b
8763:n
8759:a
8736:n
8733:=
8728:1
8725:n
8703:1
8700:=
8695:n
8691:1
8677:b
8659:n
8655:b
8649:n
8645:a
8622:1
8616:b
8596:1
8590:b
8568:n
8564:b
8541:n
8537:a
8526:x
8505:n
8501:b
8495:n
8491:a
8485:=
8482:x
8469:b
8465:a
8449:b
8446:a
8421:n
8417:x
8402:x
8398:n
8371:1
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8365:x
8362:=
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8294:n
8290:θ
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8272:r
8268:1
8265:r
8261:1
8258:r
8254:n
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8232:n
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8213:n
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8168:n
8164:/
8150:n
8146:n
8142:n
8138:n
8122:.
8119:a
8115:/
8111:b
8108:=
8079:,
8076:r
8072:/
8068:b
8065:=
8037:,
8034:r
8030:/
8026:a
8023:=
7970:2
7966:b
7962:+
7957:2
7953:a
7947:=
7944:r
7926:r
7910:.
7905:n
7901:/
7894:i
7890:e
7880:n
7876:r
7871:=
7865:n
7855:i
7851:e
7847:r
7832:n
7812:.
7806:4
7802:2
7797:i
7784:,
7778:4
7774:2
7765:,
7759:4
7755:2
7750:i
7746:,
7740:4
7736:2
7722:n
7718:ω
7714:ω
7710:ω
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7675:n
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7656:,
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7639:,
7629:,
7614:n
7610:n
7601:π
7589:z
7583:z
7570:z
7566:r
7551:z
7539:n
7522:i
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7475:/
7468:2
7442:.
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7433:n
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7420:(
7410:i
7407:+
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7349:=
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7317:1
7311:n
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7295:,
7290:2
7281:,
7274:,
7271:1
7259:n
7255:n
7210:z
7186:z
7169:π
7165:θ
7161:π
7159:−
7154:π
7150:θ
7118:2
7114:/
7107:i
7103:e
7094:r
7089:=
7079:i
7075:e
7071:r
7041:.
7036:2
7032:/
7025:i
7021:e
7012:r
7004:=
6994:i
6990:e
6986:r
6956:.
6953:)
6950:i
6947:+
6944:1
6941:(
6934:2
6930:1
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6860:2
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6811:n
6807:n
6783:|
6781:r
6777:r
6772:r
6758:,
6754:|
6750:x
6746:|
6742:=
6737:n
6732:|
6727:r
6723:|
6712:r
6708:n
6704:x
6697:b
6693:b
6677:.
6672:x
6664:b
6654:n
6651:1
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6641:=
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6606:.
6601:n
6597:x
6589:b
6578:=
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6550:x
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6534:=
6531:r
6523:b
6515:n
6499:r
6495:x
6481:,
6478:x
6475:=
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6439:n
6424:4
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6416:4
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6219:(
6215:.
6208:.
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6132:.
6130:x
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6097:.
6085:c
6079:y
6059:x
6036:1
6033:=
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6024:0
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6000:=
5997:p
5973:p
5960:x
5956:p
5951:.
5948:c
5930:n
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5867:i
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5857:x
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5843:)
5840:i
5837:,
5834:n
5831:(
5828:P
5823:i
5813:1
5807:n
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5799:=
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5768:=
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5762:1
5759:,
5756:4
5753:(
5750:P
5730:n
5710:i
5690:)
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5665:n
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5600:)
5597:x
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5591:p
5585:(
5582:x
5566:.
5554:4
5551:=
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5545:1
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5539:4
5536:(
5533:P
5499:.
5463:+
5460:x
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5445:y
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5216:n
5213:(
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5197:1
5191:n
5187:x
5183:n
5171:y
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5157:x
5154:=
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5140:+
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5131:x
5124:=
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5114:z
5100:n
5088:5
5085:x
5079:4
5076:x
5064:5
5061:x
5053:4
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4987:A
4983:n
4959:.
4952:1
4946:n
4941:k
4937:x
4933:1
4925:n
4922:A
4917:+
4912:k
4908:x
4901:n
4897:1
4891:n
4885:=
4880:1
4877:+
4874:k
4870:x
4840:1
4834:n
4829:k
4825:x
4821:n
4816:A
4808:n
4803:k
4799:x
4787:k
4783:x
4779:=
4774:1
4771:+
4768:k
4764:x
4747:0
4744:x
4734:A
4729:n
4698:1
4691:|
4687:x
4683:|
4658:n
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4641:t
4637:!
4634:n
4629:)
4626:t
4623:k
4617:s
4614:(
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4560:=
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4551:s
4546:)
4542:x
4539:+
4536:1
4533:(
4509:n
4505:q
4501:p
4484:n
4480:q
4455:n
4451:p
4426:n
4422:q
4416:/
4409:n
4405:p
4400:=
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4390:r
4378:q
4374:p
4360:q
4356:/
4352:p
4349:=
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4322:2
4317:+
4314:1
4311:=
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4299:)
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4288:+
4285:1
4281:(
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4261:2
4255:+
4250:2
4245:2
4242:+
4237:2
4233:1
4227:=
4222:2
4219:+
4214:2
4209:2
4206:+
4203:1
4198:=
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4186:2
4183:+
4180:3
4154:.
4149:2
4144:+
4141:1
4138:=
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4126:2
4123:+
4120:3
4079:.
4073:b
4070:+
4067:a
4059:3
4053:2
4049:b
4043:+
4037:3
4032:b
4029:a
4017:3
4011:2
4007:a
3998:=
3991:)
3984:3
3978:2
3974:b
3968:+
3962:3
3957:b
3954:a
3942:3
3936:2
3932:a
3925:(
3920:)
3913:3
3909:b
3904:+
3898:3
3894:a
3888:(
3879:3
3873:2
3869:b
3863:+
3857:3
3852:b
3849:a
3837:3
3831:2
3827:a
3818:=
3809:3
3805:b
3800:+
3794:3
3790:a
3784:1
3745:5
3742:4
3737:=
3732:5
3721:4
3715:=
3709:5
3704:5
3692:5
3685:2
3680:4
3674:=
3668:5
3661:2
3656:4
3626:5
3619:2
3614:4
3608:=
3602:5
3599:2
3593:4
3566:5
3563:2
3557:4
3554:=
3548:5
3545:2
3529:=
3523:5
3519:2
3506:=
3500:5
3469:5
3465:/
3416:n
3411:b
3408:a
3402:=
3396:n
3392:b
3381:n
3377:a
3347:=
3342:2
3338:i
3334:=
3331:i
3325:i
3322:=
3317:1
3304:1
3273:,
3270:1
3267:=
3262:1
3253:1
3240:1
3227:1
3206:n
3182:n
3178:b
3172:n
3168:a
3162:=
3152:n
3147:b
3144:a
3131:n
3127:b
3119:n
3115:a
3110:=
3100:n
3095:b
3092:a
3063:b
3043:a
3033:n
3017:.
3012:m
3008:)
3001:n
2997:a
2992:(
2989:=
2984:m
2980:)
2974:n
2970:/
2966:1
2962:a
2958:(
2955:=
2950:n
2946:/
2942:m
2938:a
2934:=
2929:n
2925:/
2921:1
2917:)
2911:m
2907:a
2903:(
2900:=
2894:n
2888:m
2884:a
2855:a
2833:n
2829:/
2825:1
2821:x
2810:n
2772:=
2762:3
2757:8
2744:2
2741:=
2731:3
2727:8
2696:3
2692:x
2680:x
2664:.
2661:x
2658:=
2653:3
2649:r
2636:x
2628:r
2624:x
2614:.
2599:3
2595:x
2590:=
2587:y
2554:i
2549:i
2545:i
2522:=
2487:.
2484:x
2481:=
2476:2
2472:r
2459:x
2451:r
2447:x
2437:.
2423:x
2415:=
2412:y
2368:b
2348:a
2326:b
2318:a
2294:a
2274:,
2269:a
2244:r
2224:n
2201:n
2197:r
2168:(
2148:(
2127:(
2114:n
2106:n
2084:=
2079:2
2061:n
2057:n
2053:n
2046:n
2042:x
2038:n
2034:n
2026:x
2000:=
1994:5
1989:2
1973:n
1969:x
1965:n
1961:n
1957:n
1953:n
1946:x
1938:n
1934:n
1930:n
1913:n
1909:x
1895:n
1889:n
1885:x
1866:.
1863:x
1860:=
1855:n
1851:r
1838:x
1834:n
1830:r
1826:n
1822:n
1818:x
1812:n
1775:n
1737:n
1712:e
1705:t
1698:v
1646:=
1642:)
1634:(
1560:=
1524:n
1479:=
1474:}
1355:{
1329:=
1324:}
1206:=
1201:}
1091:=
1086:}
976:=
971:}
957:+
936:+
915:+
894:+
848:e
841:t
834:v
769:x
765:n
746:n
742:x
716:x
712:n
708:n
701:0
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689:n
685:n
670:i
652:x
647:i
618:x
613:i
610:+
594:x
589:x
587:−
583:n
566:n
562:x
550:x
534:x
522:x
506:.
501:n
497:/
493:1
489:x
485:=
479:n
475:x
449:n
431:x
417:n
399:x
365:n
361:x
349:x
345:n
334:9
322:9
318:3
307:n
275:x
263:n
247:.
244:x
241:=
231:n
221:r
209:r
203:r
196:=
191:n
187:r
176:x
172:n
161:r
157:x
148:n
132:)
126:(
121:)
117:(
107:·
100:·
93:·
86:·
59:.
34:.
20:)
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