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#1050;#1086;#1088;#1085;#1080; #1090;#1072;#1082;#1078;#1077; #1084;#1086;#1078;#1085;#1086; #1086;#1087;#1088;#1077;#1076;#1077;#1083;#1080;#1090;#1100; #1082;#1072;#1082; #1095;#1072;#1089;#1090;#1085;#1099;#1077; #1089;#1083;#1091;#1095;#1072;#1080; #1074;#1086;#1079;#1074;#1077;#1076;#1077;#1085;#1080;#1103; #1074; #1089;#1090;#1077;#1087;#1077;#1085;#1100;, #1075;#1076;#1077; #1087;#1086;#1082;#1072;#1079;#1072;#1090;#1077;#1083;#1100; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1076;#1088;#1086;#1073;#1100;#1102; :

${\ displaystyle {\ sqrt [{n}] {x}} = x ^ {1 / n}.}$

#1040;#1088;#1080;#1092;#1084;#1077;#1090;#1080;#1095;#1077;#1089;#1082;#1080;#1077; #1086;#1087;#1077;#1088;#1072;#1094;#1080;#1080; v #1090; #1077;

#1044;#1086;#1087;#1086;#1083;#1085;#1077;#1085;#1080;#1077; (+) ${\ displaystyle \ scriptstyle \ left. {\ begin {matrix} \ scriptstyle {\ text {term}} \, + \, {\ text {term}} \\\ scriptstyle {\ text {summand}} \, + \, {\ text {summand}} \\\ scriptstyle {\ text {addend}} \, + \, {\ text {addend}} \\\ scriptstyle {\ text {augend}} \, + \, {\ text {addend}} \ end {matrix}} \ right \} \, = \,}$ ${\ displaystyle \ scriptstyle {\ text {sum}}}$ #1042;#1099;#1095;#1080;#1090;#1072;#1085;#1080;#1077; (-) ${\ displaystyle \ scriptstyle \ left. {\ begin {matrix} \ scriptstyle {\ text {term}} \, - \, {\ text {term}} \\\ scriptstyle {\ text {minuend}} \, - \, {\ text {subtrahend}} \ end {matrix}} \ right \} \, = \,}$ ${\ displaystyle \ scriptstyle {\ text {#1088;#1072;#1079;#1085;#1080;#1094;#1072;}}}$ #1059;#1084;#1085;#1086;#1078;#1077;#1085;#1080;#1077; (times;) ${\ displaystyle \ scriptstyle \ left. {\ begin {matrix} \ scriptstyle {\ text {factor}} \, \ times \, {\ text {factor}} \\\ scriptstyle {\ text {multiplier}} \, \ #1088;#1072;#1079; \, {\ text {multiplicand}} \ end {matrix}} \ right \} \, = \,}$ ${\ displaystyle \ scriptstyle {\ text {product}}}$ #1044;#1077;#1083;#1077;#1085;#1080;#1077; (divide;) ${\ displaystyle \ scriptstyle \ left. {\ begin {matrix} \ scriptstyle {\ frac {\ scriptstyle {\ text {divisor}}} {\ scriptstyle {\ text {divisor}}}} \\\ scriptstyle {\ text { }} \\\ scriptstyle {\ frac {\ scriptstyle {\ text {numerator}}} {\ scriptstyle {\ text {denominator}}}} \ end {matrix}} \ right \} \, = \,}$ ${\ displaystyle {\ begin {matrix} \ scriptstyle {\ text {#1076;#1088;#1086;#1073;#1100;}} \\\ scriptstyle {\ text {quotient}} \\\ scriptstyle {\ text {ratio}} \ end {matrix}}}$ #1042;#1086;#1079;#1074;#1077;#1076;#1077;#1085;#1080;#1077; #1074; #1089;#1090;#1077;#1087;#1077;#1085;#1100; ${\ displaystyle \ scriptstyle {\ text {base}} ^ {\ text {exponent}} \, = \,}$ ${\ displaystyle \ scriptstyle {\ text {power}}}$ #1082;#1086;#1088;#1077;#1085;#1100; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; (radic;) ${\ displaystyle \ scriptstyle {\ sqrt [{\ text {#1089;#1090;#1077;#1087;#1077;#1085;#1100;}}] {\ scriptstyle {\ text {radicand}}}} \, = \,}$ ${\ displaystyle \ scriptstyle {\ text {root}}}$ #1051;#1086;#1075;#1072;#1088;#1080;#1092;#1084; (#1078;#1091;#1088;#1085;#1072;#1083;) ${\ displaystyle \ scriptstyle \ log _ {\ text {base}} ({\ text {#1072;#1085;#1090;#1080;#1083;#1086;#1075;#1072;#1088;#1080;#1092;#1084;}}) \, = \,}$ ${\ displaystyle \ scriptstyle {\ text {#1083;#1086;#1075;#1072;#1088;#1080;#1092;#1084;}}}$

#1050;#1086;#1088;#1085;#1080; #1080;#1089;#1087;#1086;#1083;#1100;#1079;#1091;#1102;#1090;#1089;#1103; #1076;#1083;#1103; #1086;#1087;#1088;#1077;#1076;#1077;#1083;#1077;#1085;#1080;#1103; #1088;#1072;#1076;#1080;#1091;#1089;#1072; #1089;#1093;#1086;#1076;#1080;#1084;#1086;#1089;#1090;#1080; #1074; #1074;#1080;#1076;#1077; #1089;#1090;#1077;#1087;#1077;#1085;#1085;#1086;#1075;#1086; #1088;#1103;#1076;#1072; #1089; #1080;#1089;#1087;#1099;#1090;#1072;#1085;#1080;#1077;#1084; #1082;#1086;#1088;#1085;#1103;. #1042; #1087; - #1077; #1082;#1086;#1088;#1085;#1080; 1 #1085;#1072;#1079;#1099;#1074;#1072;#1102;#1090;#1089;#1103; #1082;#1086;#1088;#1085;#1103;#1084;#1080; #1080;#1079; #1077;#1076;#1080;#1085;#1080;#1094;#1099; #1080; #1080;#1075;#1088;#1072;#1102;#1090; #1092;#1091;#1085;#1076;#1072;#1084;#1077;#1085;#1090;#1072;#1083;#1100;#1085;#1091;#1102; #1088;#1086;#1083;#1100; #1074; #1088;#1072;#1079;#1083;#1080;#1095;#1085;#1099;#1093; #1086;#1073;#1083;#1072;#1089;#1090;#1103;#1093; #1084;#1072;#1090;#1077;#1084;#1072;#1090;#1080;#1082;#1080;, #1090;#1072;#1082;#1080;#1077; #1082;#1072;#1082; #1090;#1077;#1086;#1088;#1080;#1103; #1095;#1080;#1089;#1077;#1083;, #1090;#1077;#1086;#1088;#1080;#1103; #1091;#1088;#1072;#1074;#1085;#1077;#1085;#1080;#1081; #1080; #1087;#1088;#1077;#1086;#1073;#1088;#1072;#1079;#1086;#1074;#1072;#1085;#1080;#1077; #1060;#1091;#1088;#1100;#1077;.

#1057;#1054;#1044;#1045;#1056;#1046;#1040;#1053;#1048;#1045;

• 1 #1048;#1089;#1090;#1086;#1088;#1080;#1103;
• 2 #1054;#1087;#1088;#1077;#1076;#1077;#1083;#1077;#1085;#1080;#1077; #1080; #1086;#1073;#1086;#1079;#1085;#1072;#1095;#1077;#1085;#1080;#1103;
  • 2.1 #1050;#1074;#1072;#1076;#1088;#1072;#1090;#1085;#1099;#1077; #1082;#1086;#1088;#1085;#1080;
  • 2.2 #1050;#1091;#1073;#1080;#1095;#1077;#1089;#1082;#1080;#1077; #1082;#1086;#1088;#1085;#1080;
• 3 #1048;#1076;#1077;#1085;#1090;#1080;#1095;#1085;#1086;#1089;#1090;#1080; #1080; #1089;#1074;#1086;#1081;#1089;#1090;#1074;#1072;
• 4 #1059;#1087;#1088;#1086;#1097;#1077;#1085;#1085;#1072;#1103; #1092;#1086;#1088;#1084;#1072; #1088;#1072;#1076;#1080;#1082;#1072;#1083;#1100;#1085;#1086;#1075;#1086; #1074;#1099;#1088;#1072;#1078;#1077;#1085;#1080;#1103;
• 5 #1041;#1077;#1089;#1082;#1086;#1085;#1077;#1095;#1085;#1072;#1103; #1089;#1077;#1088;#1080;#1103;
• 6 #1042;#1099;#1095;#1080;#1089;#1083;#1077;#1085;#1080;#1077; #1075;#1083;#1072;#1074;#1085;#1099;#1093; #1082;#1086;#1088;#1085;#1077;#1081;
  • 6.1 #1048;#1089;#1087;#1086;#1083;#1100;#1079;#1086;#1074;#1072;#1085;#1080;#1077; #1084;#1077;#1090;#1086;#1076;#1072; #1053;#1100;#1102;#1090;#1086;#1085;#1072;
  • 6.2 #1055;#1086;#1089;#1083;#1077;#1076;#1086;#1074;#1072;#1090;#1077;#1083;#1100;#1085;#1086;#1077; #1074;#1099;#1095;#1080;#1089;#1083;#1077;#1085;#1080;#1077; #1075;#1083;#1072;#1074;#1085;#1099;#1093; #1082;#1086;#1088;#1085;#1077;#1081; #1076;#1077;#1089;#1103;#1090;#1080;#1095;#1085;#1099;#1093; #1095;#1080;#1089;#1077;#1083; (#1089; #1086;#1089;#1085;#1086;#1074;#1072;#1085;#1080;#1077;#1084; 10).
    • 6.2.1 #1055;#1088;#1080;#1084;#1077;#1088;#1099;
  • 6.3 #1051;#1086;#1075;#1072;#1088;#1080;#1092;#1084;#1080;#1095;#1077;#1089;#1082;#1080;#1081; #1088;#1072;#1089;#1095;#1077;#1090;
• 7 #1043;#1077;#1086;#1084;#1077;#1090;#1088;#1080;#1095;#1077;#1089;#1082;#1072;#1103; #1082;#1086;#1085;#1089;#1090;#1088;#1091;#1082;#1090;#1080;#1074;#1085;#1086;#1089;#1090;#1100;
• 8 #1089;#1083;#1086;#1078;#1085;#1099;#1093; #1082;#1086;#1088;#1085;#1077;#1081;
  • 8.1 #1050;#1074;#1072;#1076;#1088;#1072;#1090;#1085;#1099;#1077; #1082;#1086;#1088;#1085;#1080;
  • 8.2 #1050;#1086;#1088;#1085;#1080; #1077;#1076;#1080;#1085;#1089;#1090;#1074;#1072;
  • 8.3 #1082;#1086;#1088;#1085;#1080; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080;
• 9 #1056;#1077;#1096;#1077;#1085;#1080;#1077; #1084;#1085;#1086;#1075;#1086;#1095;#1083;#1077;#1085;#1086;#1074;
• 10 #1044;#1086;#1082;#1072;#1079;#1072;#1090;#1077;#1083;#1100;#1089;#1090;#1074;#1086; #1080;#1088;#1088;#1072;#1094;#1080;#1086;#1085;#1072;#1083;#1100;#1085;#1086;#1089;#1090;#1080; #1085;#1077;#1089;#1086;#1074;#1077;#1088;#1096;#1077;#1085;#1085;#1086;#1075;#1086; #1087; - #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1093;
• 11 #1057;#1084;. #1058;#1072;#1082;#1078;#1077;
• 12 #1057;#1089;#1099;#1083;#1082;#1080;
• 13 #1042;#1085;#1077;#1096;#1085;#1080;#1077; #1089;#1089;#1099;#1083;#1082;#1080;

#1048;#1089;#1090;#1086;#1088;#1080;#1103;

#1054;#1089;#1085;#1086;#1074;#1085;#1099;#1077; #1089;#1090;#1072;#1090;#1100;#1080;: #1050;#1074;#1072;#1076;#1088;#1072;#1090;#1085;#1099;#1081; #1082;#1086;#1088;#1077;#1085;#1100; sect; #1048;#1089;#1090;#1086;#1088;#1080;#1103; #1080; #1050;#1091;#1073;#1080;#1095;#1077;#1089;#1082;#1080;#1081; #1082;#1086;#1088;#1077;#1085;#1100; sect; #1048;#1089;#1090;#1086;#1088;#1080;#1103;

#1040;#1088;#1093;#1072;#1080;#1095;#1085;#1099;#1081; #1090;#1077;#1088;#1084;#1080;#1085; #1076;#1083;#1103; #1080;#1079;#1074;#1083;#1077;#1095;#1077;#1085;#1080;#1103; #1082;#1086;#1088;#1085;#1077;#1081; n - #1101;#1090;#1086; #1088;#1072;#1076;#1080;#1082;#1072;#1083;#1080;#1079;#1072;#1094;#1080;#1103;.

#1054;#1087;#1088;#1077;#1076;#1077;#1083;#1077;#1085;#1080;#1077; #1080; #1086;#1073;#1086;#1079;#1085;#1072;#1095;#1077;#1085;#1080;#1103;

#1063;#1077;#1090;#1099;#1088;#1077; #1082;#1086;#1088;#1085;#1103; #1095;#1077;#1090;#1074;#1077;#1088;#1090;#1086;#1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1080;#1079; minus;1, #1085;#1080; #1086;#1076;#1080;#1085; #1080;#1079; #1082;#1086;#1090;#1086;#1088;#1099;#1093; #1085;#1077; #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1076;#1077;#1081;#1089;#1090;#1074;#1080;#1090;#1077;#1083;#1100;#1085;#1099;#1084; #1058;#1088;#1080; #1082;#1086;#1088;#1085;#1103; #1090;#1088;#1077;#1090;#1100;#1077;#1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1080;#1079; minus;1, #1086;#1076;#1080;#1085; #1080;#1079; #1082;#1086;#1090;#1086;#1088;#1099;#1093; #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1086;#1090;#1088;#1080;#1094;#1072;#1090;#1077;#1083;#1100;#1085;#1099;#1084; #1074;#1077;#1097;#1077;#1089;#1090;#1074;#1077;#1085;#1085;#1099;#1084; #1095;#1080;#1089;#1083;#1086;#1084;.

#1055; - #1081; #1082;#1086;#1088;#1077;#1085;#1100; #1080;#1079; #1095;#1080;#1089;#1083;#1072; #1093;, #1075;#1076;#1077; #1087; #1087;#1088;#1077;#1076;#1089;#1090;#1072;#1074;#1083;#1103;#1077;#1090; #1089;#1086;#1073;#1086;#1081; #1087;#1086;#1083;#1086;#1078;#1080;#1090;#1077;#1083;#1100;#1085;#1086;#1077; #1094;#1077;#1083;#1086;#1077; #1095;#1080;#1089;#1083;#1086;, #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1083;#1102;#1073;#1099;#1084; #1080;#1079; #1087; #1076;#1077;#1081;#1089;#1090;#1074;#1080;#1090;#1077;#1083;#1100;#1085;#1099;#1093; #1080;#1083;#1080; #1082;#1086;#1084;#1087;#1083;#1077;#1082;#1089;#1085;#1099;#1093; #1095;#1080;#1089;#1077;#1083; #1075; #1082;#1086;#1090;#1086;#1088;#1086;#1075;#1086; #1087; - #1081; #1084;#1086;#1097;#1085;#1086;#1089;#1090;#1080; #1093;:

${\ displaystyle r ^ {n} = x.}$

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${\ displaystyle {\ sqrt {25}} = 5.}$

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${\ displaystyle r ^ {3} = x.}$

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${\ displaystyle {\ sqrt [{3}] {8}} = 2}$ #1072; #1090;#1072;#1082;#1078;#1077; ${\ displaystyle {\ sqrt [{3}] {- 8}} = - 2.}$

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${\ displaystyle {\ sqrt [{n}] {a ^ {m}}} = (a ^ {m}) ^ {1 / n} = a ^ {m / n} = (a ^ {1 / n}) ^ {m} = ({\ sqrt [{n}] {a}}) ^ {m}.}$

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${\ displaystyle {\ begin {align} {\ sqrt [{n}] {ab}} amp; = {\ sqrt [{n}] {a}} {\ sqrt [{n}] {b}} \\ { \ sqrt [{n}] {\ frac {a} {b}}} amp; = {\ frac {\ sqrt [{n}] {a}} {\ sqrt [{n}] {b}}} \ end {#1074;#1099;#1088;#1086;#1074;#1085;#1077;#1085;#1086;}}}$

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${\ displaystyle {\ sqrt {-1}} \ times {\ sqrt {-1}} \ neq {\ sqrt {-1 \ times -1}} = 1, \ quad}$ #1089;#1082;#1086;#1088;#1077;#1077;, ${\ displaystyle \ quad {\ sqrt {-1}} \ times {\ sqrt {-1}} = i \ times i = i ^ {2} = - 1.}$

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3. #1042; #1079;#1085;#1072;#1084;#1077;#1085;#1072;#1090;#1077;#1083;#1077; #1085;#1077;#1090; #1088;#1072;#1076;#1080;#1082;#1072;#1083;#1086;#1074;.

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${\ displaystyle {\ sqrt {\ tfrac {32} {5}}} = {\ sqrt {\ tfrac {16 \ times 2} {5}}} = 4 {\ sqrt {\ tfrac {2} {5}} }}$

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${\ displaystyle 4 {\ sqrt {\ tfrac {2} {5}}} = {\ frac {4 {\ sqrt {2}}} {\ sqrt {5}}}}$

#1053;#1072;#1082;#1086;#1085;#1077;#1094;, #1091;#1076;#1072;#1083;#1103;#1077;#1084; #1088;#1072;#1076;#1080;#1082;#1072;#1083; #1080;#1079; #1079;#1085;#1072;#1084;#1077;#1085;#1072;#1090;#1077;#1083;#1103; #1089;#1083;#1077;#1076;#1091;#1102;#1097;#1080;#1084; #1086;#1073;#1088;#1072;#1079;#1086;#1084;:

${\ displaystyle {\ frac {4 {\ sqrt {2}}} {\ sqrt {5}}} = {\ frac {4 {\ sqrt {2}}} {\ sqrt {5}}} \ cdot {\ frac {\ sqrt {5}} {\ sqrt {5}}} = {\ frac {4 {\ sqrt {10}}} {5}} = {\ frac {4} {5}} {\ sqrt {10 }}}$

#1050;#1086;#1075;#1076;#1072; #1077;#1089;#1090;#1100; #1079;#1085;#1072;#1084;#1077;#1085;#1072;#1090;#1077;#1083;#1100;, #1074;#1082;#1083;#1102;#1095;#1072;#1102;#1097;#1080;#1081; #1089;#1091;#1088;#1076;#1099;, #1074;#1089;#1077;#1075;#1076;#1072; #1084;#1086;#1078;#1085;#1086; #1085;#1072;#1081;#1090;#1080; #1084;#1085;#1086;#1078;#1080;#1090;#1077;#1083;#1100;, #1085;#1072; #1082;#1086;#1090;#1086;#1088;#1099;#1081; #1084;#1086;#1078;#1085;#1086; #1091;#1084;#1085;#1086;#1078;#1080;#1090;#1100; #1095;#1080;#1089;#1083;#1080;#1090;#1077;#1083;#1100; #1080; #1079;#1085;#1072;#1084;#1077;#1085;#1072;#1090;#1077;#1083;#1100;, #1095;#1090;#1086;#1073;#1099; #1091;#1087;#1088;#1086;#1089;#1090;#1080;#1090;#1100; #1074;#1099;#1088;#1072;#1078;#1077;#1085;#1080;#1077;. #1053;#1072;#1087;#1088;#1080;#1084;#1077;#1088;, #1080;#1089;#1087;#1086;#1083;#1100;#1079;#1091;#1103; #1092;#1072;#1082;#1090;#1086;#1088;#1080;#1079;#1072;#1094;#1080;#1102; #1089;#1091;#1084;#1084;#1099; #1076;#1074;#1091;#1093; #1082;#1091;#1073;#1086;#1074; :

${\ displaystyle {\ frac {1} {{\ sqrt [{3}] {a}} + {\ sqrt [{3}] {b}}}} = {\ frac {{\ sqrt [{3}]) {a ^ {2}}} - {\ sqrt [{3}] {ab}} + {\ sqrt [{3}] {b ^ {2}}}} {\ left ({\ sqrt [{3} ] {a}} + {\ sqrt [{3}] {b}} \ right) \ left ({\ sqrt [{3}] {a ^ {2}}} - {\ sqrt [{3}] { ab}} + {\ sqrt [{3}] {b ^ {2}}} \ right)}} = {\ frac {{\ sqrt [{3}] {a ^ {2}}} - {\ sqrt [{3}] {ab}} + {\ sqrt [{3}] {b ^ {2}}}} {a + b}}.}.$

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${\ displaystyle {\ sqrt {3 + 2 {\ sqrt {2}}}} = 1 + {\ sqrt {2}}}$

#1042;#1099;#1096;#1077;#1091;#1087;#1086;#1084;#1103;#1085;#1091;#1090;#1086;#1077; #1084;#1086;#1078;#1085;#1086; #1087;#1086;#1083;#1091;#1095;#1080;#1090;#1100; #1089; #1087;#1086;#1084;#1086;#1097;#1100;#1102;:

${\ displaystyle {\ sqrt {3 + 2 {\ sqrt {2}}}} = {\ sqrt {1 + 2 {\ sqrt {2}} + 2}} = {\ sqrt {1 ^ {2} +2 {\ sqrt {2}} + {\ sqrt {2}} ^ {2}}} = {\ sqrt {\ left (1 + {\ sqrt {2}} \ right) ^ {2}}} = 1+ {\ sqrt {2}}}$

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${\ displaystyle (1 + x) ^ {\ frac {s} {t}} = \ sum _ {n = 0} ^ {\ infty} {\ frac {\ prod _ {k = 0} ^ {n-1 } (s-kt)} {n! t ^ {n}}} x ^ {n}}$

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${\ displaystyle x_ {k + 1} = {\ frac {n-1} {n}} x_ {k} + {\ frac {A} {nx_ {k} ^ {n-1}}}}$

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${\ displaystyle {\ sqrt [{n}] {z}} = {\ sqrt [{n}] {x ^ {n} + y}} = x + {\ cfrac {y} {nx ^ {n-1} + {\ cfrac {(n-1) y} {2x + {\ cfrac {(n + 1) y} {3nx ^ {n-1} + {\ cfrac {(2n-1) y} {2x + {\ cfrac) {(2n + 1) y} {5nx ^ {n-1} + {\ cfrac {(3n-1) y} {2x + \ ddots}}}}}}}}}}}}.}.}$

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#1055;#1088;#1080;#1084;#1077;#1088;#1099;

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   1 2. 3 4  / \/ 01 52.27 56

   01     100middot;1middot;00middot;12 + 101middot;2middot;01middot;11  le;  1 lt; 100middot;1middot;00middot;22 + 101middot;2middot;01middot;21   x = 1 01      y = 100middot;1middot;00middot;12 + 101middot;2middot;01middot;12 = 1 + 0 =  1 00 52    100middot;1middot;10middot;22 + 101middot;2middot;11middot;21  le;  52 lt; 100middot;1middot;10middot;32 + 101middot;2middot;11middot;31   x = 2 00 44     y = 100middot;1middot;10middot;22 + 101middot;2middot;11middot;21 = 4 + 40 = 44 08 27    100middot;1middot;120middot;32 + 101middot;2middot;121middot;31 le; 827 lt; 100middot;1middot;120middot;42 + 101middot;2middot;121middot;41  x = 3 07 29     y = 100middot;1middot;120middot;32 + 101middot;2middot;121middot;31 = 9 + 720 = 729 98 56   100middot;1middot;1230middot;42 + 101middot;2middot;1231middot;41 le; 9856 lt; 100middot;1middot;1230middot;52 + 101middot;2middot;1231middot;51  x = 4 98 56    y = 100middot;1middot;1230middot;42 + 101middot;2middot;1231middot;41 = 16 + 9840 = 9856 00 00   Algorithm terminates: Answer is 12.34

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   1 6. 1 2 4 3 / \/ 004 192.000 000 000

  004      100middot;1middot;00middot;13 + 101middot;3middot;01middot;12 + 102middot;3middot;02middot;11 le;   4 lt; 100middot;1middot;00middot;23  + 101middot;3middot;01middot;22 + 102middot;3middot;02middot;21  x = 1 001       y = 100middot;1middot;00middot;13 + 101middot;3middot;01middot;12 + 102middot;3middot;02middot;11 = 1 +  0 +   0 =   1 003 192     100middot;1middot;10middot;63 + 101middot;3middot;11middot;62 + 102middot;3middot;12middot;61 le;  3192 lt; 100middot;1middot;10middot;73  + 101middot;3middot;11middot;72 + 102middot;3middot;12middot;71  x = 6 003 096      y = 100middot;1middot;10middot;63 + 101middot;3middot;11middot;62 + 102middot;3middot;12middot;61 = 216 + 1,080 +  1,800 =  3,096 096 000    100middot;1middot;160middot;13 + 101middot;3middot;161middot;12 + 102middot;3middot;162middot;11 le;  96000 lt; 100middot;1middot;160middot;23 + 101middot;3middot;161middot;22 + 102middot;3middot;162middot;21 x = 1 077 281     y = 100middot;1middot;160middot;13 + 101middot;3middot;161middot;12 + 102middot;3middot;162middot;11 = 1 + 480 +  76,800 =  77,281 018 719 000   100middot;1middot;1610middot;23 + 101middot;3middot;1611middot;22 + 102middot;3middot;1612middot;21 le; 18719000 lt; 100middot;1middot;1610middot;33 + 101middot;3middot;1611middot;32 + 102middot;3middot;1612middot;31 x = 2 015 571 928   y = 100middot;1middot;1610middot;23 + 101middot;3middot;1611middot;22 + 102middot;3middot;1612middot;21 = 8 + 19,320 + 15,552,600 = 15,571,928 003 147 072 000 100middot;1middot;16120middot;43 + 101middot;3middot;16121middot;42 + 102middot;3middot;16122middot;41 le; 3147072000 lt; 100middot;1middot;16120middot;53 + 101middot;3middot;16121middot;52 + 102middot;3middot;16122middot;51 x = 4 The desired precision is achieved: The cube root of 4192 is about 16.12

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#1063;#1080;#1089;#1083;#1086; 1 #1080;#1084;#1077;#1077;#1090; n #1088;#1072;#1079;#1083;#1080;#1095;#1085;#1099;#1093; #1082;#1086;#1088;#1085;#1077;#1081; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1085;#1072; #1082;#1086;#1084;#1087;#1083;#1077;#1082;#1089;#1085;#1086;#1081; #1087;#1083;#1086;#1089;#1082;#1086;#1089;#1090;#1080;, #1072; #1080;#1084;#1077;#1085;#1085;#1086;

${\ displaystyle 1, \; \ omega, \; \ omega ^ {2}, \; \ ldots, \; \ omega ^ {n-1},}$

#1082;#1091;#1076;#1072;

${\ displaystyle \ omega = e ^ {\ frac {2 \ pi i} {n}} = \ cos \ left ({\ frac {2 \ pi} {n}} \ right) + i \ sin \ left ({ \ frac {2 \ pi} {n}} \ right)}$

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n- #1099;#1077; #1082;#1086;#1088;#1085;#1080;

#1043;#1077;#1086;#1084;#1077;#1090;#1088;#1080;#1095;#1077;#1089;#1082;#1086;#1077; #1087;#1088;#1077;#1076;#1089;#1090;#1072;#1074;#1083;#1077;#1085;#1080;#1077; #1082;#1086;#1088;#1085;#1077;#1081; 2ndash;6 #1082;#1086;#1084;#1087;#1083;#1077;#1082;#1089;#1085;#1086;#1075;#1086; #1095;#1080;#1089;#1083;#1072; z #1074; #1087;#1086;#1083;#1103;#1088;#1085;#1086;#1081; #1092;#1086;#1088;#1084;#1077; re ^iphi;,thinsp; #1075;#1076;#1077; r = | z thinsp;| #1080; phi; = arg z. #1045;#1089;#1083;#1080; z #1074;#1077;#1097;#1077;#1089;#1090;#1074;#1077;#1085;#1085;#1086;#1077; #1095;#1080;#1089;#1083;#1086;, phi; = 0 #1080;#1083;#1080; pi;. #1054;#1089;#1085;#1086;#1074;#1085;#1099;#1077; #1082;#1086;#1088;#1085;#1080; #1087;#1086;#1082;#1072;#1079;#1072;#1085;#1099; #1095;#1077;#1088;#1085;#1099;#1084;.

#1050;#1072;#1078;#1076;#1086;#1077; #1082;#1086;#1084;#1087;#1083;#1077;#1082;#1089;#1085;#1086;#1077; #1095;#1080;#1089;#1083;#1086; #1080;#1084;#1077;#1077;#1090; n #1088;#1072;#1079;#1083;#1080;#1095;#1085;#1099;#1093; #1082;#1086;#1088;#1085;#1077;#1081; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1085;#1072; #1082;#1086;#1084;#1087;#1083;#1077;#1082;#1089;#1085;#1086;#1081; #1087;#1083;#1086;#1089;#1082;#1086;#1089;#1090;#1080;. #1069;#1090;#1080;

${\ displaystyle \ eta, \; \ eta \ omega, \; \ eta \ omega ^ {2}, \; \ ldots, \; \ eta \ omega ^ {n-1},}$

#1075;#1076;#1077; eta; - #1077;#1076;#1080;#1085;#1089;#1090;#1074;#1077;#1085;#1085;#1099;#1081; #1082;#1086;#1088;#1077;#1085;#1100; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080;, #1072; 1, omega;, omega; ²,... omega; ^{n minus;1} - #1082;#1086;#1088;#1085;#1080; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1080;#1079; #1077;#1076;#1080;#1085;#1080;#1094;#1099;. #1053;#1072;#1087;#1088;#1080;#1084;#1077;#1088;, #1095;#1077;#1090;#1099;#1088;#1077; #1088;#1072;#1079;#1085;#1099;#1093; #1082;#1086;#1088;#1085;#1103; #1095;#1077;#1090;#1074;#1077;#1088;#1090;#1086;#1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1080;#1079; 2 #1088;#1072;#1074;#1085;#1099;

${\ displaystyle {\ sqrt [{4}] {2}}, \ quad i {\ sqrt [{4}] {2}}, \ quad - {\ sqrt [{4}] {2}}, \ quad {\ text {and}} \ quad -i {\ sqrt [{4}] {2}}.}$

#1042; #1087;#1086;#1083;#1103;#1088;#1085;#1086;#1081; #1092;#1086;#1088;#1084;#1077; #1077;#1076;#1080;#1085;#1089;#1090;#1074;#1077;#1085;#1085;#1099;#1081; #1082;#1086;#1088;#1077;#1085;#1100; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1084;#1086;#1078;#1077;#1090; #1073;#1099;#1090;#1100; #1085;#1072;#1081;#1076;#1077;#1085; #1087;#1086; #1092;#1086;#1088;#1084;#1091;#1083;#1077;

${\ displaystyle {\ sqrt [{n}] {re ^ {i \ theta}}} = {\ sqrt [{n}] {r}} \ cdot e ^ {i \ theta / n}.}$

#1047;#1076;#1077;#1089;#1100; r - #1074;#1077;#1083;#1080;#1095;#1080;#1085;#1072; (#1084;#1086;#1076;#1091;#1083;#1100;, #1090;#1072;#1082;#1078;#1077; #1085;#1072;#1079;#1099;#1074;#1072;#1077;#1084;#1099;#1081; #1072;#1073;#1089;#1086;#1083;#1102;#1090;#1085;#1099;#1084; #1079;#1085;#1072;#1095;#1077;#1085;#1080;#1077;#1084; ) #1095;#1080;#1089;#1083;#1072;, #1080;#1079; #1082;#1086;#1090;#1086;#1088;#1086;#1075;#1086; #1089;#1083;#1077;#1076;#1091;#1077;#1090; #1080;#1079;#1074;#1083;#1077;#1095;#1100; #1082;#1086;#1088;#1077;#1085;#1100;; #1077;#1089;#1083;#1080; #1095;#1080;#1089;#1083;#1086; #1084;#1086;#1078;#1085;#1086; #1079;#1072;#1087;#1080;#1089;#1072;#1090;#1100; #1082;#1072;#1082; + bi, #1090;#1086;#1075;#1076;#1072;. #1050;#1088;#1086;#1084;#1077; #1090;#1086;#1075;#1086;, #1101;#1090;#1086; #1091;#1075;#1086;#1083;, #1086;#1073;#1088;#1072;#1079;#1086;#1074;#1072;#1085;#1085;#1099;#1081; #1087;#1088;#1080; #1087;#1086;#1074;#1086;#1088;#1086;#1090;#1077; #1074; #1080;#1089;#1093;#1086;#1076;#1085;#1086;#1081; #1090;#1086;#1095;#1082;#1077; #1087;#1088;#1086;#1090;#1080;#1074; #1095;#1072;#1089;#1086;#1074;#1086;#1081; #1089;#1090;#1088;#1077;#1083;#1082;#1080; #1086;#1090; #1087;#1086;#1083;#1086;#1078;#1080;#1090;#1077;#1083;#1100;#1085;#1086;#1081; #1075;#1086;#1088;#1080;#1079;#1086;#1085;#1090;#1072;#1083;#1100;#1085;#1086;#1081; #1086;#1089;#1080; #1082; #1083;#1091;#1095;#1091;, #1080;#1076;#1091;#1097;#1077;#1084;#1091; #1086;#1090; #1085;#1072;#1095;#1072;#1083;#1072; #1082;#1086;#1086;#1088;#1076;#1080;#1085;#1072;#1090; #1082; #1095;#1080;#1089;#1083;#1091;; #1086;#1085; #1080;#1084;#1077;#1077;#1090; #1089;#1074;#1086;#1081;#1089;#1090;#1074;#1072;, #1082;#1086;#1090;#1086;#1088;#1099;#1077; #1080;${\ displaystyle r = {\ sqrt {a ^ {2} + b ^ {2}}}}$${\ displaystyle \ theta}$${\ Displaystyle \ #1089;#1086;#1079; \ #1090;#1077;#1090;#1072; = #1072; / #1075;,}$ ${\ Displaystyle \ #1075;#1088;#1077;#1093; \ #1090;#1077;#1090;#1072; = #1073; / #1075;,}$${\ displaystyle \ tan \ theta = b / a.}$

#1058;#1072;#1082;#1080;#1084; #1086;#1073;#1088;#1072;#1079;#1086;#1084;, #1087;#1086;#1080;#1089;#1082; #1082;#1086;#1088;#1085;#1077;#1081; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080; #1074; #1082;#1086;#1084;#1087;#1083;#1077;#1082;#1089;#1085;#1086;#1081; #1087;#1083;#1086;#1089;#1082;#1086;#1089;#1090;#1080; #1084;#1086;#1078;#1085;#1086; #1088;#1072;#1079;#1076;#1077;#1083;#1080;#1090;#1100; #1085;#1072; #1076;#1074;#1072; #1101;#1090;#1072;#1087;#1072;. #1042;#1086;- #1087;#1077;#1088;#1074;#1099;#1093;, #1074;#1077;#1083;#1080;#1095;#1080;#1085;#1072; #1074;#1089;#1077;#1093; #1087; - #1081; #1082;#1086;#1088;#1085;#1077;#1081; #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1087; - #1081; #1082;#1086;#1088;#1077;#1085;#1100; #1080;#1079; #1074;#1077;#1083;#1080;#1095;#1080;#1085;#1099; #1080;#1089;#1093;#1086;#1076;#1085;#1086;#1075;#1086; #1095;#1080;#1089;#1083;#1072;. #1042;#1086;-#1074;#1090;#1086;#1088;#1099;#1093;, #1091;#1075;#1086;#1083; #1084;#1077;#1078;#1076;#1091; #1087;#1086;#1083;#1086;#1078;#1080;#1090;#1077;#1083;#1100;#1085;#1086;#1081; #1075;#1086;#1088;#1080;#1079;#1086;#1085;#1090;#1072;#1083;#1100;#1085;#1086;#1081; #1086;#1089;#1100;#1102; #1080; #1083;#1091;#1095;#1086;#1084; #1086;#1090; #1085;#1072;#1095;#1072;#1083;#1072; #1082;#1086;#1086;#1088;#1076;#1080;#1085;#1072;#1090; #1076;#1086; #1086;#1076;#1085;#1086;#1075;#1086; #1080;#1079; #1082;#1086;#1088;#1085;#1077;#1081; n- #1075;#1086; #1087;#1086;#1088;#1103;#1076;#1082;#1072; #1088;#1072;#1074;#1077;#1085;, #1075;#1076;#1077; - #1091;#1075;#1086;#1083;, #1086;#1087;#1088;#1077;#1076;#1077;#1083;#1077;#1085;#1085;#1099;#1081; #1090;#1072;#1082;#1080;#1084; #1078;#1077; #1086;#1073;#1088;#1072;#1079;#1086;#1084; #1076;#1083;#1103; #1095;#1080;#1089;#1083;#1072;, #1080;#1079; #1082;#1086;#1090;#1086;#1088;#1086;#1075;#1086; #1080;#1079;#1074;#1083;#1077;#1082;#1072;#1077;#1090;#1089;#1103; #1082;#1086;#1088;#1077;#1085;#1100;. #1050;#1088;#1086;#1084;#1077; #1090;#1086;#1075;#1086;, #1074;#1089;#1077; #1082;#1086;#1088;#1085;#1080; n #1080;#1079; n #1088;#1072;#1089;#1087;#1086;#1083;#1086;#1078;#1077;#1085;#1099; #1085;#1072; #1086;#1076;#1080;#1085;#1072;#1082;#1086;#1074;#1086;#1084; #1088;#1072;#1089;#1089;#1090;#1086;#1103;#1085;#1080;#1080; #1076;#1088;#1091;#1075; #1086;#1090; #1076;#1088;#1091;#1075;#1072;. ${\ displaystyle \ theta / n}$${\ displaystyle \ theta}$

#1045;#1089;#1083;#1080; n #1095;#1077;#1090;#1085;#1086;, n- #1077; #1082;#1086;#1088;#1085;#1080; #1082;#1086;#1084;#1087;#1083;#1077;#1082;#1089;#1085;#1086;#1075;#1086; #1095;#1080;#1089;#1083;#1072;, #1080;#1079; #1082;#1086;#1090;#1086;#1088;#1099;#1093; #1077;#1089;#1090;#1100; #1095;#1077;#1090;#1085;#1086;#1077; #1095;#1080;#1089;#1083;#1086;, #1074;#1093;#1086;#1076;#1103;#1090; #1074; #1072;#1076;#1076;#1080;#1090;#1080;#1074;#1085;#1099;#1077; #1086;#1073;#1088;#1072;#1090;#1085;#1099;#1077; #1087;#1072;#1088;#1099;, #1090;#1072;#1082; #1095;#1090;#1086; #1077;#1089;#1083;#1080; #1095;#1080;#1089;#1083;#1086; r 1 #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1086;#1076;#1085;#1080;#1084; #1080;#1079; #1082;#1086;#1088;#1085;#1077;#1081; n- #1081; #1089;#1090;#1077;#1087;#1077;#1085;#1080;, #1090;#1086; r 2 = - r 1 #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1076;#1088;#1091;#1075;#1080;#1084;. #1069;#1090;#1086; #1089;#1074;#1103;#1079;#1072;#1085;#1086; #1089; #1090;#1077;#1084;, #1095;#1090;#1086; #1087;#1086;#1074;#1099;#1096;#1077;#1085;#1080;#1077; #1082;#1086;#1101;#1092;#1092;#1080;#1094;#1080;#1077;#1085;#1090;#1072; ndash;1 #1087;#1086;#1089;#1083;#1077;#1076;#1085;#1077;#1075;#1086; #1076;#1086; #1089;#1090;#1077;#1087;#1077;#1085;#1080; n #1076;#1083;#1103; #1095;#1077;#1090;#1085;#1086;#1075;#1086; n #1076;#1072;#1077;#1090; 1: #1090;#1086; #1077;#1089;#1090;#1100; (- r 1) ⁿ = (ndash;1) ⁿ times; r 1 ⁿ = r 1 ⁿ.

#1050;#1072;#1082; #1080; #1074; #1089;#1083;#1091;#1095;#1072;#1077; #1089; #1082;#1074;#1072;#1076;#1088;#1072;#1090;#1085;#1099;#1084;#1080; #1082;#1086;#1088;#1085;#1103;#1084;#1080;, #1087;#1088;#1080;#1074;#1077;#1076;#1077;#1085;#1085;#1072;#1103; #1074;#1099;#1096;#1077; #1092;#1086;#1088;#1084;#1091;#1083;#1072; #1085;#1077; #1086;#1087;#1088;#1077;#1076;#1077;#1083;#1103;#1077;#1090; #1085;#1077;#1087;#1088;#1077;#1088;#1099;#1074;#1085;#1091;#1102; #1092;#1091;#1085;#1082;#1094;#1080;#1102; #1085;#1072; #1074;#1089;#1077;#1081; #1082;#1086;#1084;#1087;#1083;#1077;#1082;#1089;#1085;#1086;#1081; #1087;#1083;#1086;#1089;#1082;#1086;#1089;#1090;#1080;, #1072; #1074;#1084;#1077;#1089;#1090;#1086; #1101;#1090;#1086;#1075;#1086; #1080;#1084;#1077;#1077;#1090; #1088;#1072;#1079;#1074;#1077;#1090;#1074;#1083;#1077;#1085;#1080;#1077; #1074; #1090;#1086;#1095;#1082;#1072;#1093;, #1075;#1076;#1077; theta; / n #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1088;#1072;#1079;#1088;#1099;#1074;#1085;#1099;#1084;.

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