[{"data":1,"prerenderedAt":2497},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fbrueche-multiplizieren":546,"formula-related-de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fbrueche-multiplizieren":2487},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":555,"category":65,"checks":2433,"conditions":2453,"conditionsHtml":2454,"deprecated":2455,"description":2427,"extension":2456,"figure":2455,"figureCaption":2455,"figureHtml":2455,"formulaId":2457,"latex":2458,"latexHtml":2459,"level":2460,"meta":2461,"navigation":2462,"path":2463,"redirectFrom":2464,"related":2465,"seo":2469,"stem":2470,"subject":585,"summary":2471,"tags":2472,"variables":2477,"__hash__":2486},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Farithmetic\u002Fbrueche-multiplizieren.md","Brüche multiplizieren",[550,551,552,553,554],"Bruch mal Bruch","Brüche malnehmen","Bruch mal ganze Zahl","Zähler mal Zähler","kreuzweise kürzen",{"type":556,"value":557,"toc":2426},"minimark",[558,563,572,772,779,782,786,1150,1153,1526,1703,2089,2093],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,571],"p",{},"Beim Multiplizieren braucht man ",[568,569,570],"strong",{},"keinen gemeinsamen Nenner",". Man multipliziert die Zähler miteinander und die Nenner miteinander.",[564,573,574,575,620,621,771],{},"Eine ganze Zahl schreibt man dafür als Bruch mit dem Nenner ",[576,577,580,602],"span",{"className":578},[579],"katex",[576,581,584],{"className":582},[583],"katex-mathml",[585,586,588],"math",{"xmlns":587},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[589,590,591,598],"semantics",{},[592,593,594],"mrow",{},[595,596,597],"mn",{},"1",[599,600,597],"annotation",{"encoding":601},"application\u002Fx-tex",[576,603,607],{"className":604,"ariaHidden":606},[605],"katex-html","true",[576,608,611,616],{"className":609},[610],"base",[576,612],{"className":613,"style":615},[614],"strut","height:0.6444em;",[576,617,597],{"className":618},[619],"mord",", zum Beispiel ",[576,622,624,650],{"className":623},[579],[576,625,627],{"className":626},[583],[585,628,629],{"xmlns":587},[589,630,631,647],{},[592,632,633,636,640],{},[595,634,635],{},"3",[637,638,639],"mo",{},"=",[641,642,643,645],"mfrac",{},[595,644,635],{},[595,646,597],{},[599,648,649],{"encoding":601},"3 = \\frac{3}{1}",[576,651,653,674],{"className":652,"ariaHidden":606},[605],[576,654,656,659,662,667,671],{"className":655},[610],[576,657],{"className":658,"style":615},[614],[576,660,635],{"className":661},[619],[576,663],{"className":664,"style":666},[665],"mspace","margin-right:0.2778em;",[576,668,639],{"className":669},[670],"mrel",[576,672],{"className":673,"style":666},[665],[576,675,677,681],{"className":676},[610],[576,678],{"className":679,"style":680},[614],"height:1.1901em;vertical-align:-0.345em;",[576,682,684,689,767],{"className":683},[619],[576,685],{"className":686},[687,688],"mopen","nulldelimiter",[576,690,692],{"className":691},[641],[576,693,697,758],{"className":694},[695,696],"vlist-t","vlist-t2",[576,698,701,753],{"className":699},[700],"vlist-r",[576,702,706,727,738],{"className":703,"style":705},[704],"vlist","height:0.8451em;",[576,707,709,714],{"style":708},"top:-2.655em;",[576,710],{"className":711,"style":713},[712],"pstrut","height:3em;",[576,715,721],{"className":716},[717,718,719,720],"sizing","reset-size6","size3","mtight",[576,722,724],{"className":723},[619,720],[576,725,597],{"className":726},[619,720],[576,728,730,733],{"style":729},"top:-3.23em;",[576,731],{"className":732,"style":713},[712],[576,734],{"className":735,"style":737},[736],"frac-line","border-bottom-width:0.04em;",[576,739,741,744],{"style":740},"top:-3.394em;",[576,742],{"className":743,"style":713},[712],[576,745,747],{"className":746},[717,718,719,720],[576,748,750],{"className":749},[619,720],[576,751,635],{"className":752},[619,720],[576,754,757],{"className":755},[756],"vlist-s","​",[576,759,761],{"className":760},[700],[576,762,765],{"className":763,"style":764},[704],"height:0.345em;",[576,766],{},[576,768],{"className":769},[770,688],"mclose",". Dann wird nur der Zähler mit der Zahl multipliziert.",[564,773,774,775,778],{},"Es lohnt sich, ",[568,776,777],{},"vor dem Ausrechnen zu kürzen",": Ein Zähler darf mit einem Nenner gekürzt werden, auch wenn sie zu verschiedenen Brüchen gehören („kreuzweise kürzen\"). So bleiben die Zahlen klein.",[564,780,781],{},"Die Nenner dürfen nicht null sein.",[559,783,785],{"id":784},"beispiele","Beispiele",[576,787,790],{"className":788},[789],"katex-display",[576,791,793,841],{"className":792},[579],[576,794,796],{"className":795},[583],[585,797,799],{"xmlns":587,"display":798},"block",[589,800,801,838],{},[592,802,803,810,813,820,822,830,832],{},[641,804,805,808],{},[595,806,807],{},"2",[595,809,635],{},[637,811,812],{},"⋅",[641,814,815,817],{},[595,816,635],{},[595,818,819],{},"4",[637,821,639],{},[641,823,824,827],{},[595,825,826],{},"6",[595,828,829],{},"12",[637,831,639],{},[641,833,834,836],{},[595,835,597],{},[595,837,807],{},[599,839,840],{"encoding":601},"\\frac{2}{3} \\cdot \\frac{3}{4} = \\frac{6}{12} = \\frac{1}{2}",[576,842,844,928,1005,1082],{"className":843,"ariaHidden":606},[605],[576,845,847,851,917,921,925],{"className":846},[610],[576,848],{"className":849,"style":850},[614],"height:2.0074em;vertical-align:-0.686em;",[576,852,854,857,914],{"className":853},[619],[576,855],{"className":856},[687,688],[576,858,860],{"className":859},[641],[576,861,863,905],{"className":862},[695,696],[576,864,866,902],{"className":865},[700],[576,867,870,882,890],{"className":868,"style":869},[704],"height:1.3214em;",[576,871,873,876],{"style":872},"top:-2.314em;",[576,874],{"className":875,"style":713},[712],[576,877,879],{"className":878},[619],[576,880,635],{"className":881},[619],[576,883,884,887],{"style":729},[576,885],{"className":886,"style":713},[712],[576,888],{"className":889,"style":737},[736],[576,891,893,896],{"style":892},"top:-3.677em;",[576,894],{"className":895,"style":713},[712],[576,897,899],{"className":898},[619],[576,900,807],{"className":901},[619],[576,903,757],{"className":904},[756],[576,906,908],{"className":907},[700],[576,909,912],{"className":910,"style":911},[704],"height:0.686em;",[576,913],{},[576,915],{"className":916},[770,688],[576,918],{"className":919,"style":920},[665],"margin-right:0.2222em;",[576,922,812],{"className":923},[924],"mbin",[576,926],{"className":927,"style":920},[665],[576,929,931,934,996,999,1002],{"className":930},[610],[576,932],{"className":933,"style":850},[614],[576,935,937,940,993],{"className":936},[619],[576,938],{"className":939},[687,688],[576,941,943],{"className":942},[641],[576,944,946,985],{"className":945},[695,696],[576,947,949,982],{"className":948},[700],[576,950,952,963,971],{"className":951,"style":869},[704],[576,953,954,957],{"style":872},[576,955],{"className":956,"style":713},[712],[576,958,960],{"className":959},[619],[576,961,819],{"className":962},[619],[576,964,965,968],{"style":729},[576,966],{"className":967,"style":713},[712],[576,969],{"className":970,"style":737},[736],[576,972,973,976],{"style":892},[576,974],{"className":975,"style":713},[712],[576,977,979],{"className":978},[619],[576,980,635],{"className":981},[619],[576,983,757],{"className":984},[756],[576,986,988],{"className":987},[700],[576,989,991],{"className":990,"style":911},[704],[576,992],{},[576,994],{"className":995},[770,688],[576,997],{"className":998,"style":666},[665],[576,1000,639],{"className":1001},[670],[576,1003],{"className":1004,"style":666},[665],[576,1006,1008,1011,1073,1076,1079],{"className":1007},[610],[576,1009],{"className":1010,"style":850},[614],[576,1012,1014,1017,1070],{"className":1013},[619],[576,1015],{"className":1016},[687,688],[576,1018,1020],{"className":1019},[641],[576,1021,1023,1062],{"className":1022},[695,696],[576,1024,1026,1059],{"className":1025},[700],[576,1027,1029,1040,1048],{"className":1028,"style":869},[704],[576,1030,1031,1034],{"style":872},[576,1032],{"className":1033,"style":713},[712],[576,1035,1037],{"className":1036},[619],[576,1038,829],{"className":1039},[619],[576,1041,1042,1045],{"style":729},[576,1043],{"className":1044,"style":713},[712],[576,1046],{"className":1047,"style":737},[736],[576,1049,1050,1053],{"style":892},[576,1051],{"className":1052,"style":713},[712],[576,1054,1056],{"className":1055},[619],[576,1057,826],{"className":1058},[619],[576,1060,757],{"className":1061},[756],[576,1063,1065],{"className":1064},[700],[576,1066,1068],{"className":1067,"style":911},[704],[576,1069],{},[576,1071],{"className":1072},[770,688],[576,1074],{"className":1075,"style":666},[665],[576,1077,639],{"className":1078},[670],[576,1080],{"className":1081,"style":666},[665],[576,1083,1085,1088],{"className":1084},[610],[576,1086],{"className":1087,"style":850},[614],[576,1089,1091,1094,1147],{"className":1090},[619],[576,1092],{"className":1093},[687,688],[576,1095,1097],{"className":1096},[641],[576,1098,1100,1139],{"className":1099},[695,696],[576,1101,1103,1136],{"className":1102},[700],[576,1104,1106,1117,1125],{"className":1105,"style":869},[704],[576,1107,1108,1111],{"style":872},[576,1109],{"className":1110,"style":713},[712],[576,1112,1114],{"className":1113},[619],[576,1115,807],{"className":1116},[619],[576,1118,1119,1122],{"style":729},[576,1120],{"className":1121,"style":713},[712],[576,1123],{"className":1124,"style":737},[736],[576,1126,1127,1130],{"style":892},[576,1128],{"className":1129,"style":713},[712],[576,1131,1133],{"className":1132},[619],[576,1134,597],{"className":1135},[619],[576,1137,757],{"className":1138},[756],[576,1140,1142],{"className":1141},[700],[576,1143,1145],{"className":1144,"style":911},[704],[576,1146],{},[576,1148],{"className":1149},[770,688],[564,1151,1152],{},"Ganze Zahl mal Bruch:",[576,1154,1156],{"className":1155},[789],[576,1157,1159,1206],{"className":1158},[579],[576,1160,1162],{"className":1161},[583],[585,1163,1164],{"xmlns":587,"display":798},[589,1165,1166,1203],{},[592,1167,1168,1170,1172,1179,1181,1187,1189,1195,1197],{},[595,1169,635],{},[637,1171,812],{},[641,1173,1174,1176],{},[595,1175,807],{},[595,1177,1178],{},"5",[637,1180,639],{},[641,1182,1183,1185],{},[595,1184,635],{},[595,1186,597],{},[637,1188,812],{},[641,1190,1191,1193],{},[595,1192,807],{},[595,1194,1178],{},[637,1196,639],{},[641,1198,1199,1201],{},[595,1200,826],{},[595,1202,1178],{},[599,1204,1205],{"encoding":601},"3 \\cdot \\frac{2}{5} = \\frac{3}{1} \\cdot \\frac{2}{5} = \\frac{6}{5}",[576,1207,1209,1227,1304,1381,1458],{"className":1208,"ariaHidden":606},[605],[576,1210,1212,1215,1218,1221,1224],{"className":1211},[610],[576,1213],{"className":1214,"style":615},[614],[576,1216,635],{"className":1217},[619],[576,1219],{"className":1220,"style":920},[665],[576,1222,812],{"className":1223},[924],[576,1225],{"className":1226,"style":920},[665],[576,1228,1230,1233,1295,1298,1301],{"className":1229},[610],[576,1231],{"className":1232,"style":850},[614],[576,1234,1236,1239,1292],{"className":1235},[619],[576,1237],{"className":1238},[687,688],[576,1240,1242],{"className":1241},[641],[576,1243,1245,1284],{"className":1244},[695,696],[576,1246,1248,1281],{"className":1247},[700],[576,1249,1251,1262,1270],{"className":1250,"style":869},[704],[576,1252,1253,1256],{"style":872},[576,1254],{"className":1255,"style":713},[712],[576,1257,1259],{"className":1258},[619],[576,1260,1178],{"className":1261},[619],[576,1263,1264,1267],{"style":729},[576,1265],{"className":1266,"style":713},[712],[576,1268],{"className":1269,"style":737},[736],[576,1271,1272,1275],{"style":892},[576,1273],{"className":1274,"style":713},[712],[576,1276,1278],{"className":1277},[619],[576,1279,807],{"className":1280},[619],[576,1282,757],{"className":1283},[756],[576,1285,1287],{"className":1286},[700],[576,1288,1290],{"className":1289,"style":911},[704],[576,1291],{},[576,1293],{"className":1294},[770,688],[576,1296],{"className":1297,"style":666},[665],[576,1299,639],{"className":1300},[670],[576,1302],{"className":1303,"style":666},[665],[576,1305,1307,1310,1372,1375,1378],{"className":1306},[610],[576,1308],{"className":1309,"style":850},[614],[576,1311,1313,1316,1369],{"className":1312},[619],[576,1314],{"className":1315},[687,688],[576,1317,1319],{"className":1318},[641],[576,1320,1322,1361],{"className":1321},[695,696],[576,1323,1325,1358],{"className":1324},[700],[576,1326,1328,1339,1347],{"className":1327,"style":869},[704],[576,1329,1330,1333],{"style":872},[576,1331],{"className":1332,"style":713},[712],[576,1334,1336],{"className":1335},[619],[576,1337,597],{"className":1338},[619],[576,1340,1341,1344],{"style":729},[576,1342],{"className":1343,"style":713},[712],[576,1345],{"className":1346,"style":737},[736],[576,1348,1349,1352],{"style":892},[576,1350],{"className":1351,"style":713},[712],[576,1353,1355],{"className":1354},[619],[576,1356,635],{"className":1357},[619],[576,1359,757],{"className":1360},[756],[576,1362,1364],{"className":1363},[700],[576,1365,1367],{"className":1366,"style":911},[704],[576,1368],{},[576,1370],{"className":1371},[770,688],[576,1373],{"className":1374,"style":920},[665],[576,1376,812],{"className":1377},[924],[576,1379],{"className":1380,"style":920},[665],[576,1382,1384,1387,1449,1452,1455],{"className":1383},[610],[576,1385],{"className":1386,"style":850},[614],[576,1388,1390,1393,1446],{"className":1389},[619],[576,1391],{"className":1392},[687,688],[576,1394,1396],{"className":1395},[641],[576,1397,1399,1438],{"className":1398},[695,696],[576,1400,1402,1435],{"className":1401},[700],[576,1403,1405,1416,1424],{"className":1404,"style":869},[704],[576,1406,1407,1410],{"style":872},[576,1408],{"className":1409,"style":713},[712],[576,1411,1413],{"className":1412},[619],[576,1414,1178],{"className":1415},[619],[576,1417,1418,1421],{"style":729},[576,1419],{"className":1420,"style":713},[712],[576,1422],{"className":1423,"style":737},[736],[576,1425,1426,1429],{"style":892},[576,1427],{"className":1428,"style":713},[712],[576,1430,1432],{"className":1431},[619],[576,1433,807],{"className":1434},[619],[576,1436,757],{"className":1437},[756],[576,1439,1441],{"className":1440},[700],[576,1442,1444],{"className":1443,"style":911},[704],[576,1445],{},[576,1447],{"className":1448},[770,688],[576,1450],{"className":1451,"style":666},[665],[576,1453,639],{"className":1454},[670],[576,1456],{"className":1457,"style":666},[665],[576,1459,1461,1464],{"className":1460},[610],[576,1462],{"className":1463,"style":850},[614],[576,1465,1467,1470,1523],{"className":1466},[619],[576,1468],{"className":1469},[687,688],[576,1471,1473],{"className":1472},[641],[576,1474,1476,1515],{"className":1475},[695,696],[576,1477,1479,1512],{"className":1478},[700],[576,1480,1482,1493,1501],{"className":1481,"style":869},[704],[576,1483,1484,1487],{"style":872},[576,1485],{"className":1486,"style":713},[712],[576,1488,1490],{"className":1489},[619],[576,1491,1178],{"className":1492},[619],[576,1494,1495,1498],{"style":729},[576,1496],{"className":1497,"style":713},[712],[576,1499],{"className":1500,"style":737},[736],[576,1502,1503,1506],{"style":892},[576,1504],{"className":1505,"style":713},[712],[576,1507,1509],{"className":1508},[619],[576,1510,826],{"className":1511},[619],[576,1513,757],{"className":1514},[756],[576,1516,1518],{"className":1517},[700],[576,1519,1521],{"className":1520,"style":911},[704],[576,1522],{},[576,1524],{"className":1525},[770,688],[564,1527,1528,1529,1557,1558,1587,1588,1616,1617,1557,1645,1587,1674,1702],{},"Vorher kreuzweise kürzen: ",[576,1530,1532,1545],{"className":1531},[579],[576,1533,1535],{"className":1534},[583],[585,1536,1537],{"xmlns":587},[589,1538,1539,1543],{},[592,1540,1541],{},[595,1542,819],{},[599,1544,819],{"encoding":601},[576,1546,1548],{"className":1547,"ariaHidden":606},[605],[576,1549,1551,1554],{"className":1550},[610],[576,1552],{"className":1553,"style":615},[614],[576,1555,819],{"className":1556},[619]," und ",[576,1559,1561,1575],{"className":1560},[579],[576,1562,1564],{"className":1563},[583],[585,1565,1566],{"xmlns":587},[589,1567,1568,1573],{},[592,1569,1570],{},[595,1571,1572],{},"8",[599,1574,1572],{"encoding":601},[576,1576,1578],{"className":1577,"ariaHidden":606},[605],[576,1579,1581,1584],{"className":1580},[610],[576,1582],{"className":1583,"style":615},[614],[576,1585,1572],{"className":1586},[619]," durch ",[576,1589,1591,1604],{"className":1590},[579],[576,1592,1594],{"className":1593},[583],[585,1595,1596],{"xmlns":587},[589,1597,1598,1602],{},[592,1599,1600],{},[595,1601,819],{},[599,1603,819],{"encoding":601},[576,1605,1607],{"className":1606,"ariaHidden":606},[605],[576,1608,1610,1613],{"className":1609},[610],[576,1611],{"className":1612,"style":615},[614],[576,1614,819],{"className":1615},[619],", ",[576,1618,1620,1633],{"className":1619},[579],[576,1621,1623],{"className":1622},[583],[585,1624,1625],{"xmlns":587},[589,1626,1627,1631],{},[592,1628,1629],{},[595,1630,635],{},[599,1632,635],{"encoding":601},[576,1634,1636],{"className":1635,"ariaHidden":606},[605],[576,1637,1639,1642],{"className":1638},[610],[576,1640],{"className":1641,"style":615},[614],[576,1643,635],{"className":1644},[619],[576,1646,1648,1662],{"className":1647},[579],[576,1649,1651],{"className":1650},[583],[585,1652,1653],{"xmlns":587},[589,1654,1655,1660],{},[592,1656,1657],{},[595,1658,1659],{},"9",[599,1661,1659],{"encoding":601},[576,1663,1665],{"className":1664,"ariaHidden":606},[605],[576,1666,1668,1671],{"className":1667},[610],[576,1669],{"className":1670,"style":615},[614],[576,1672,1659],{"className":1673},[619],[576,1675,1677,1690],{"className":1676},[579],[576,1678,1680],{"className":1679},[583],[585,1681,1682],{"xmlns":587},[589,1683,1684,1688],{},[592,1685,1686],{},[595,1687,635],{},[599,1689,635],{"encoding":601},[576,1691,1693],{"className":1692,"ariaHidden":606},[605],[576,1694,1696,1699],{"className":1695},[610],[576,1697],{"className":1698,"style":615},[614],[576,1700,635],{"className":1701},[619],":",[576,1704,1706],{"className":1705},[789],[576,1707,1709,1763],{"className":1708},[579],[576,1710,1712],{"className":1711},[583],[585,1713,1714],{"xmlns":587,"display":798},[589,1715,1716,1760],{},[592,1717,1718,1724,1726,1732,1734,1752,1754],{},[641,1719,1720,1722],{},[595,1721,819],{},[595,1723,1659],{},[637,1725,812],{},[641,1727,1728,1730],{},[595,1729,635],{},[595,1731,1572],{},[637,1733,639],{},[641,1735,1736,1744],{},[592,1737,1738,1740,1742],{},[595,1739,597],{},[637,1741,812],{},[595,1743,597],{},[592,1745,1746,1748,1750],{},[595,1747,635],{},[637,1749,812],{},[595,1751,807],{},[637,1753,639],{},[641,1755,1756,1758],{},[595,1757,597],{},[595,1759,826],{},[599,1761,1762],{"encoding":601},"\\frac{4}{9} \\cdot \\frac{3}{8} = \\frac{1 \\cdot 1}{3 \\cdot 2} = \\frac{1}{6}",[576,1764,1766,1843,1920,2021],{"className":1765,"ariaHidden":606},[605],[576,1767,1769,1772,1834,1837,1840],{"className":1768},[610],[576,1770],{"className":1771,"style":850},[614],[576,1773,1775,1778,1831],{"className":1774},[619],[576,1776],{"className":1777},[687,688],[576,1779,1781],{"className":1780},[641],[576,1782,1784,1823],{"className":1783},[695,696],[576,1785,1787,1820],{"className":1786},[700],[576,1788,1790,1801,1809],{"className":1789,"style":869},[704],[576,1791,1792,1795],{"style":872},[576,1793],{"className":1794,"style":713},[712],[576,1796,1798],{"className":1797},[619],[576,1799,1659],{"className":1800},[619],[576,1802,1803,1806],{"style":729},[576,1804],{"className":1805,"style":713},[712],[576,1807],{"className":1808,"style":737},[736],[576,1810,1811,1814],{"style":892},[576,1812],{"className":1813,"style":713},[712],[576,1815,1817],{"className":1816},[619],[576,1818,819],{"className":1819},[619],[576,1821,757],{"className":1822},[756],[576,1824,1826],{"className":1825},[700],[576,1827,1829],{"className":1828,"style":911},[704],[576,1830],{},[576,1832],{"className":1833},[770,688],[576,1835],{"className":1836,"style":920},[665],[576,1838,812],{"className":1839},[924],[576,1841],{"className":1842,"style":920},[665],[576,1844,1846,1849,1911,1914,1917],{"className":1845},[610],[576,1847],{"className":1848,"style":850},[614],[576,1850,1852,1855,1908],{"className":1851},[619],[576,1853],{"className":1854},[687,688],[576,1856,1858],{"className":1857},[641],[576,1859,1861,1900],{"className":1860},[695,696],[576,1862,1864,1897],{"className":1863},[700],[576,1865,1867,1878,1886],{"className":1866,"style":869},[704],[576,1868,1869,1872],{"style":872},[576,1870],{"className":1871,"style":713},[712],[576,1873,1875],{"className":1874},[619],[576,1876,1572],{"className":1877},[619],[576,1879,1880,1883],{"style":729},[576,1881],{"className":1882,"style":713},[712],[576,1884],{"className":1885,"style":737},[736],[576,1887,1888,1891],{"style":892},[576,1889],{"className":1890,"style":713},[712],[576,1892,1894],{"className":1893},[619],[576,1895,635],{"className":1896},[619],[576,1898,757],{"className":1899},[756],[576,1901,1903],{"className":1902},[700],[576,1904,1906],{"className":1905,"style":911},[704],[576,1907],{},[576,1909],{"className":1910},[770,688],[576,1912],{"className":1913,"style":666},[665],[576,1915,639],{"className":1916},[670],[576,1918],{"className":1919,"style":666},[665],[576,1921,1923,1926,2012,2015,2018],{"className":1922},[610],[576,1924],{"className":1925,"style":850},[614],[576,1927,1929,1932,2009],{"className":1928},[619],[576,1930],{"className":1931},[687,688],[576,1933,1935],{"className":1934},[641],[576,1936,1938,2001],{"className":1937},[695,696],[576,1939,1941,1998],{"className":1940},[700],[576,1942,1944,1967,1975],{"className":1943,"style":869},[704],[576,1945,1946,1949],{"style":872},[576,1947],{"className":1948,"style":713},[712],[576,1950,1952,1955,1958,1961,1964],{"className":1951},[619],[576,1953,635],{"className":1954},[619],[576,1956],{"className":1957,"style":920},[665],[576,1959,812],{"className":1960},[924],[576,1962],{"className":1963,"style":920},[665],[576,1965,807],{"className":1966},[619],[576,1968,1969,1972],{"style":729},[576,1970],{"className":1971,"style":713},[712],[576,1973],{"className":1974,"style":737},[736],[576,1976,1977,1980],{"style":892},[576,1978],{"className":1979,"style":713},[712],[576,1981,1983,1986,1989,1992,1995],{"className":1982},[619],[576,1984,597],{"className":1985},[619],[576,1987],{"className":1988,"style":920},[665],[576,1990,812],{"className":1991},[924],[576,1993],{"className":1994,"style":920},[665],[576,1996,597],{"className":1997},[619],[576,1999,757],{"className":2000},[756],[576,2002,2004],{"className":2003},[700],[576,2005,2007],{"className":2006,"style":911},[704],[576,2008],{},[576,2010],{"className":2011},[770,688],[576,2013],{"className":2014,"style":666},[665],[576,2016,639],{"className":2017},[670],[576,2019],{"className":2020,"style":666},[665],[576,2022,2024,2027],{"className":2023},[610],[576,2025],{"className":2026,"style":850},[614],[576,2028,2030,2033,2086],{"className":2029},[619],[576,2031],{"className":2032},[687,688],[576,2034,2036],{"className":2035},[641],[576,2037,2039,2078],{"className":2038},[695,696],[576,2040,2042,2075],{"className":2041},[700],[576,2043,2045,2056,2064],{"className":2044,"style":869},[704],[576,2046,2047,2050],{"style":872},[576,2048],{"className":2049,"style":713},[712],[576,2051,2053],{"className":2052},[619],[576,2054,826],{"className":2055},[619],[576,2057,2058,2061],{"style":729},[576,2059],{"className":2060,"style":713},[712],[576,2062],{"className":2063,"style":737},[736],[576,2065,2066,2069],{"style":892},[576,2067],{"className":2068,"style":713},[712],[576,2070,2072],{"className":2071},[619],[576,2073,597],{"className":2074},[619],[576,2076,757],{"className":2077},[756],[576,2079,2081],{"className":2080},[700],[576,2082,2084],{"className":2083,"style":911},[704],[576,2085],{},[576,2087],{"className":2088},[770,688],[559,2090,2092],{"id":2091},"typische-fehler","Typische Fehler",[2094,2095,2096,2100,2423],"ul",{},[2097,2098,2099],"li",{},"Erst gleichnamig machen wie beim Addieren: Das ist unnötig und macht die Zahlen groß. Beim Multiplizieren rechnet man direkt Zähler mal Zähler und Nenner mal Nenner.",[2097,2101,2102,2103,2223,2224,2322,2323,2422],{},"Bei einer ganzen Zahl auch den Nenner multiplizieren: ",[576,2104,2106,2128],{"className":2105},[579],[576,2107,2109],{"className":2108},[583],[585,2110,2111],{"xmlns":587},[589,2112,2113,2125],{},[592,2114,2115,2117,2119],{},[595,2116,635],{},[637,2118,812],{},[641,2120,2121,2123],{},[595,2122,807],{},[595,2124,1178],{},[599,2126,2127],{"encoding":601},"3 \\cdot \\frac{2}{5}",[576,2129,2131,2149],{"className":2130,"ariaHidden":606},[605],[576,2132,2134,2137,2140,2143,2146],{"className":2133},[610],[576,2135],{"className":2136,"style":615},[614],[576,2138,635],{"className":2139},[619],[576,2141],{"className":2142,"style":920},[665],[576,2144,812],{"className":2145},[924],[576,2147],{"className":2148,"style":920},[665],[576,2150,2152,2155],{"className":2151},[610],[576,2153],{"className":2154,"style":680},[614],[576,2156,2158,2161,2220],{"className":2157},[619],[576,2159],{"className":2160},[687,688],[576,2162,2164],{"className":2163},[641],[576,2165,2167,2212],{"className":2166},[695,696],[576,2168,2170,2209],{"className":2169},[700],[576,2171,2173,2187,2195],{"className":2172,"style":705},[704],[576,2174,2175,2178],{"style":708},[576,2176],{"className":2177,"style":713},[712],[576,2179,2181],{"className":2180},[717,718,719,720],[576,2182,2184],{"className":2183},[619,720],[576,2185,1178],{"className":2186},[619,720],[576,2188,2189,2192],{"style":729},[576,2190],{"className":2191,"style":713},[712],[576,2193],{"className":2194,"style":737},[736],[576,2196,2197,2200],{"style":740},[576,2198],{"className":2199,"style":713},[712],[576,2201,2203],{"className":2202},[717,718,719,720],[576,2204,2206],{"className":2205},[619,720],[576,2207,807],{"className":2208},[619,720],[576,2210,757],{"className":2211},[756],[576,2213,2215],{"className":2214},[700],[576,2216,2218],{"className":2217,"style":764},[704],[576,2219],{},[576,2221],{"className":2222},[770,688]," ist ",[576,2225,2227,2245],{"className":2226},[579],[576,2228,2230],{"className":2229},[583],[585,2231,2232],{"xmlns":587},[589,2233,2234,2242],{},[592,2235,2236],{},[641,2237,2238,2240],{},[595,2239,826],{},[595,2241,1178],{},[599,2243,2244],{"encoding":601},"\\frac{6}{5}",[576,2246,2248],{"className":2247,"ariaHidden":606},[605],[576,2249,2251,2254],{"className":2250},[610],[576,2252],{"className":2253,"style":680},[614],[576,2255,2257,2260,2319],{"className":2256},[619],[576,2258],{"className":2259},[687,688],[576,2261,2263],{"className":2262},[641],[576,2264,2266,2311],{"className":2265},[695,696],[576,2267,2269,2308],{"className":2268},[700],[576,2270,2272,2286,2294],{"className":2271,"style":705},[704],[576,2273,2274,2277],{"style":708},[576,2275],{"className":2276,"style":713},[712],[576,2278,2280],{"className":2279},[717,718,719,720],[576,2281,2283],{"className":2282},[619,720],[576,2284,1178],{"className":2285},[619,720],[576,2287,2288,2291],{"style":729},[576,2289],{"className":2290,"style":713},[712],[576,2292],{"className":2293,"style":737},[736],[576,2295,2296,2299],{"style":740},[576,2297],{"className":2298,"style":713},[712],[576,2300,2302],{"className":2301},[717,718,719,720],[576,2303,2305],{"className":2304},[619,720],[576,2306,826],{"className":2307},[619,720],[576,2309,757],{"className":2310},[756],[576,2312,2314],{"className":2313},[700],[576,2315,2317],{"className":2316,"style":764},[704],[576,2318],{},[576,2320],{"className":2321},[770,688],", nicht ",[576,2324,2326,2345],{"className":2325},[579],[576,2327,2329],{"className":2328},[583],[585,2330,2331],{"xmlns":587},[589,2332,2333,2342],{},[592,2334,2335],{},[641,2336,2337,2339],{},[595,2338,826],{},[595,2340,2341],{},"15",[599,2343,2344],{"encoding":601},"\\frac{6}{15}",[576,2346,2348],{"className":2347,"ariaHidden":606},[605],[576,2349,2351,2354],{"className":2350},[610],[576,2352],{"className":2353,"style":680},[614],[576,2355,2357,2360,2419],{"className":2356},[619],[576,2358],{"className":2359},[687,688],[576,2361,2363],{"className":2362},[641],[576,2364,2366,2411],{"className":2365},[695,696],[576,2367,2369,2408],{"className":2368},[700],[576,2370,2372,2386,2394],{"className":2371,"style":705},[704],[576,2373,2374,2377],{"style":708},[576,2375],{"className":2376,"style":713},[712],[576,2378,2380],{"className":2379},[717,718,719,720],[576,2381,2383],{"className":2382},[619,720],[576,2384,2341],{"className":2385},[619,720],[576,2387,2388,2391],{"style":729},[576,2389],{"className":2390,"style":713},[712],[576,2392],{"className":2393,"style":737},[736],[576,2395,2396,2399],{"style":740},[576,2397],{"className":2398,"style":713},[712],[576,2400,2402],{"className":2401},[717,718,719,720],[576,2403,2405],{"className":2404},[619,720],[576,2406,826],{"className":2407},[619,720],[576,2409,757],{"className":2410},[756],[576,2412,2414],{"className":2413},[700],[576,2415,2417],{"className":2416,"style":764},[704],[576,2418],{},[576,2420],{"className":2421},[770,688],".",[2097,2424,2425],{},"Zwei Zähler miteinander kürzen: Gekürzt wird immer ein Zähler mit einem Nenner.",{"title":2427,"searchDepth":2428,"depth":2428,"links":2429},"",2,[2430,2431,2432],{"id":561,"depth":2428,"text":562},{"id":784,"depth":2428,"text":785},{"id":2091,"depth":2428,"text":2092},[2434,2437,2440,2442,2445,2448,2451],{"lhs":2435,"rhs":2436},"(a\u002Fb)*(c\u002Fd)","(a*c)\u002F(b*d)",{"lhs":2438,"rhs":2439},"(2\u002F3)*(3\u002F4)","6\u002F12",{"lhs":2439,"rhs":2441},"1\u002F2",{"lhs":2443,"rhs":2444},"3*(2\u002F5)","(3\u002F1)*(2\u002F5)",{"lhs":2446,"rhs":2447},"(3*2)\u002F(1*5)","6\u002F5",{"lhs":2449,"rhs":2450},"(4\u002F9)*(3\u002F8)","(1*1)\u002F(3*2)",{"lhs":2450,"rhs":2452},"1\u002F6","b \\neq 0,\\ d \\neq 0","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>b\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b \\neq 0,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>d\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">d \\neq 0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","fraction-multiplication","\\frac{a}{b} \\cdot \\frac{c}{d} = \\frac{a \\cdot c}{b \\cdot d}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmfrac>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmfrac>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>d\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>d\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\frac{a}{b} \\cdot \\frac{c}{d} = \\frac{a \\cdot c}{b \\cdot d}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.7936em;vertical-align:-0.686em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.1076em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.686em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.7936em;vertical-align:-0.686em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.1076em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.686em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.8075em;vertical-align:-0.686em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.1215em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.686em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","lower",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Farithmetic\u002Fbrueche-multiplizieren",[],[2466,2467,2468],"fraction-division","fraction-reducing","fraction-addition",{"title":548,"description":2427},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fbrueche-multiplizieren","Zwei Brüche werden multipliziert, indem man Zähler mal Zähler und Nenner mal Nenner rechnet.",[2473,2474,2475,2476],"bruch","bruchrechnung","multiplikation","kürzen",[2478,2482],{"symbol":2479,"meaning":2480,"symbolHtml":2481},"a, c","Zähler der beiden Brüche","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, c\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":2483,"meaning":2484,"symbolHtml":2485},"b, d","Nenner der beiden Brüche, nicht null","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>b\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>d\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b, d\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","ktL7qVewIRSwh2CqpnGN7UnOmnt2m9eATXy7hVwm9fg",[2488,2491,2494],{"stem":2489,"title":2490,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fbrueche-addieren-und-subtrahieren","Brüche addieren und subtrahieren",{"stem":2492,"title":2493,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fbrueche-dividieren","Brüche dividieren",{"stem":2495,"title":2496,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fbrueche-kuerzen-und-erweitern","Brüche kürzen und erweitern",1793824259268]