// Numbas version: exam_results_page_options
{"name": "Vereinfachung von Wurzelausdr\u00fccken", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Vereinfachung von Wurzelausdr\u00fccken", "tags": [], "metadata": {"description": "<p>This question tests the student's understanding of what is and is not a surd, and on their simplification of surds.</p>\n<p>Translated to German, minor changes to advice section.</p>\n<p>Original: https://numbas.mathcentre.ac.uk/question/22497/surds-simplification/ by Lauren Richards.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Einige einfache Aufgaben zur Vereinfachung von Ausdr&uuml;cken mit Quadratzahlen.</p>", "advice": "<h4>a)</h4>\n<p>$\\sqrt{\\var{square}}$ und $\\sqrt[3]{\\var{cube}}$ sind rational, und zwar sogar ganze Zahlen: $\\simplify{{sqrt(square)}}$ bzw. $\\var{root}$.</p>\n<p>$\\sqrt{\\var{h}}$, $\\sqrt{\\var{j}}$ und $\\sqrt{\\var{k}}$ sind irrational. Es gibt keine ganzen Zahlen $a, b$, $b\\ne 0$ so dass $\\left(\\frac ab\\right)^2=\\var{h}$ (oder $= \\var{j}$ oder $=\\var{k}$). Eine M&ouml;glichkeit, das zu sehen, ist, den Satz &uuml;ber die eindeutige Primzaktorzerlegung von ganzen Zahlen zu benutzen. Wenn $\\left(\\frac ab\\right)^2=\\var{h}$ gilt, also $a^2 = hb^2$, dann kommt in der eindeutigen Primfaktorzerlegung der linken Seite jede Primzahl mit einem geraden Exponenten vor; auf der rechten Seite ist das nicht der Fall.</p>\n<p></p>\n<p></p>\n<h4>b)</h4>\n<p>Benutzen SIe, dass $\\sqrt{a}\\times\\sqrt{b}=\\sqrt{ab}$.</p>\n<p>Wir suchen nach einer Quadratzahl, die $ab$ teilt, und schreiben das als $\\sqrt{b^2}\\times\\sqrt{a}$.</p>\n<p><em>i)</em></p>\n<p>$\\sqrt{48}$&nbsp;= $\\sqrt{16}\\times\\sqrt3$</p>\n<p>$\\sqrt{16}$ ist $4$, also ist die Antwort: $4\\sqrt3$.</p>\n<p><em>ii)</em></p>\n<p>$\\sqrt{56}$ = $\\sqrt{4}\\times\\sqrt{14}$</p>\n<p>$\\sqrt4$ ist $2$, also erhalten wir: $2\\sqrt{14}$.</p>\n<p><em>iii)</em></p>\n<p>$\\sqrt{32}$&nbsp;=&nbsp;$\\sqrt{16}\\times\\sqrt{2}$</p>\n<p>$\\sqrt{16}$ ist $4$, wir bekommen also: $4\\sqrt2$.</p>\n<p><em>iv)</em></p>\n<p>$\\sqrt{44}$ = $\\sqrt{4}\\times\\sqrt{11}$</p>\n<p>$\\sqrt4$ ist $2$, also erhalten wir: $2\\sqrt{11}$.</p>\n<p></p>\n<h4>c)</h4>\n<p>Beachten Sie, dass $\\sqrt{\\var{a}}$ und $\\sqrt{\\var{d}}$ Wurzeln aus Quadratzahlen sind.</p>\n<p>$\\sqrt{\\var{a}}$ = $\\var{sqrta}$, folglich:</p>\n<p><em>i)</em> $\\sqrt{\\var{c}}$ = $\\sqrt{\\var{a}}$ x $\\sqrt{\\var{b}}$ = $\\var{sqrta}\\sqrt{\\var{b}}$ and</p>\n<p><em>ii)</em> $\\sqrt{\\var{g}}$ = $\\sqrt{\\var{d}}$ x $\\sqrt{\\var{f}}$ = $\\var{sqrtd}\\sqrt{\\var{f}}$.</p>\n<h4></h4>\n<h4></h4>", "rulesets": {}, "extensions": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "a*b", "description": "<p>a times b</p>", "templateType": "anything"}, "doptions": {"name": "doptions", "group": "Ungrouped variables", "definition": "[ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 ]", "description": "<p>List of squared numbers from 1 to 144</p>", "templateType": "list of numbers"}, "sqrtd": {"name": "sqrtd", "group": "Ungrouped variables", "definition": "sqrt(d)", "description": "<p>square root of the 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"adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Vereinfachen Sie die Quadratwurzeln, indem Sie eine m&ouml;glichst gro&szlig;e Quadratzahl \"aus der Wurzel herausziehen\".</p>\n<p>[[0]]</p>", "gaps": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "choices": ["<p>i) $\\sqrt{48}$</p>", "<p>ii) $\\sqrt{32}$</p>", "<p>iii) $\\sqrt{56}$</p>", "<p>iv) $\\sqrt{44}$</p>"], "matrix": [["1", 0, 0, "0"], [0, "0", "0", "1"], ["0", "0", "1", 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