// Numbas version: finer_feedback_settings
{"name": "Irrationale Zahlen - Nenner rational machen", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Irrationale Zahlen - Nenner rational machen", "tags": [], "metadata": {"description": "<p>Rationalise the denominator with increasingly difficult examples involving compound denominators.&nbsp;</p>\n<p>Translated to German, made some minor changes to Advice section.</p>\n<p>Original: https://numbas.mathcentre.ac.uk/question/22555/rationalising-the-denominator-surds/ by Lauren Richards</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Schreiben Sie die folgenden Zahlen als Br&uuml;che mit einer ganzen Zahl im Nenner.</p>", "advice": "<h4>a)</h4>\n<p>Es gilt $\\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}} = \\sqrt{\\frac{a}{b}}$.</p>\n<p><em>i)</em></p>\n<p>$\\displaystyle\\frac{\\sqrt{\\simplify{{square_nums[0]}{prime_nums[0]}}}}{\\sqrt{\\var{prime_nums[0]}}} = \\sqrt{\\frac{{\\simplify{{square_nums[0]}{prime_nums[0]}}}}{{\\var{prime_nums[0]}}}} = \\sqrt{\\var{square_nums[0]}}= \\simplify{{sqrt(square_nums[0])}}$</p>\n<p><em>ii)</em></p>\n<p>$\\displaystyle\\frac{\\sqrt{\\simplify{{square_nums[1]}{prime_nums[1]}}}}{\\sqrt{\\var{prime_nums[1]}}} = \\sqrt{\\frac{{\\simplify{{square_nums[1]}{prime_nums[1]}}}}{{\\var{prime_nums[1]}}}} = \\sqrt{\\var{square_nums[1]}} = \\simplify{{sqrt(square_nums[1])}}$.</p>\n<h4>b)</h4>\n<p>Um eine ganze Zahl in den Nenner zu bekommen, benutzen wir, dass $\\displaystyle\\sqrt{a}\\cdot\\sqrt{a}=a$.</p>\n<p>Gefragt ist, den Ausdruck $\\displaystyle\\frac{\\sqrt{\\var{n_lcm}}}{\\sqrt{\\var{m_lcm}}}$ umzuschreiben als Bruch mit einer ganzen Zahl im Nenner.</p>\n<p>Der Nenner ist ${\\sqrt{\\var{m_lcm}}}$, also multiplizieren wir den Nenner mit ${\\sqrt{\\var{m_lcm}}}$.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\sqrt{\\var{n_lcm}}}{\\sqrt{\\var{m_lcm}}}\\cdot\\frac{\\sqrt{\\var{m_lcm}}}{\\sqrt{\\var{m_lcm}}} \\]</p>\n<p>Weil $\\sqrt{a}\\cdot\\sqrt{b}$ = $\\sqrt{ab}$, k&ouml;nnen wir den Z&auml;hler&nbsp; als $\\sqrt{\\var{num}}$ schreiben.</p>\n<p style=\"text-align: left;\">Der Nenner ist ${\\sqrt{\\var{m_lcm}}}\\cdot{\\sqrt{\\var{m_lcm}}} = {\\var{m_lcm}}$.</p>\n<p style=\"text-align: left;\">Also ist das Ergebnis dieser Rechnung</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\sqrt{\\var{num}}}{\\var{m_lcm}} \\]</p>\n<p>Zum Schluss k&ouml;nnen wir noch k&uuml;rzen und erhalten</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\sqrt{\\var{a[0]*b[0]}}}{\\var{b[0]}} \\]</p>\n<p style=\"text-align: left;\">(Man kann nat&uuml;rlich auch schon zu Beginn k&uuml;rzen, das ist im Grunde effizienter.)</p>\n<h4>c)</h4>\n<p>Zun&auml;chst erweitern wir den Bruch mit $\\sqrt{\\var{f}}$. Weil $\\displaystyle\\sqrt{a}\\cdot\\sqrt{a}=a$ gilt, ist der Nenner des entstehenden Bruchs $\\var{d}\\cdot\\var{f}=\\var{df}$.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{c}}{\\var{d}\\sqrt{\\var{f}}}\\cdot\\simplify[all,!simplifyFractions,!sqrtDivision]{sqrt({f})/(sqrt({f}))}={\\frac{\\simplify{{c}*sqrt({f})}}{{(\\var{d}\\cdot\\var{f})}}}=\\frac{\\simplify{{c}*sqrt({f})}}{\\var{d*f}}$.</p>\n<div data-jme-visible=\"{gcd_cdf} &gt; 1\">\n<p>Wir k&uuml;rzen noch mit $\\var{gcd_cdf}$, und erhalten als Ergebnis</p>\n<p style=\"text-align: left;\">\\[\\frac{\\simplify{{c}*sqrt({f})}}{\\var{d*f}}=\\frac{\\simplify{{c_coprime}*sqrt({f})}}{\\var{df_coprime}}\\]</p>\n</div>\n<p style=\"text-align: left;\">Die Antwort soll in der Form $\\frac{a \\sqrt{\\var{f}}}{b}$ gegeben werden, also ist $a = \\var{c_coprime}$ und $b = \\var{df_coprime}$.</p>\n<p style=\"text-align: left;\"></p>\n<h4>d)</h4>\n<p>Um auf eine ganze Zahl im Nenner zu kommen, wenn der Nenner eine Summe ist wie bei $\\displaystyle\\frac{\\var{g}}{\\sqrt{\\var{h}}+\\var{j}}$, erweitern wir mit $\\sqrt{\\var{h}}-\\var{j}$ und benutzen die dritte binomische Formel.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{g}}{\\sqrt{\\var{h}}+\\var{j}}\\cdot\\frac{(\\sqrt{\\var{h}}-\\var{j})}{(\\sqrt{\\var{h}}-\\var{j})}=\\frac{\\var{g}(\\sqrt{\\var{h}}-\\var{j})}{(\\sqrt{\\var{h}}+\\var{j})(\\sqrt{\\var{h}}-\\var{j})}\\text{.} \\]</p>\n<p>Wir erhalten</p>\n<p style=\"text-align: center;\">\\[ \\frac{ \\simplify{ {g*-j} + {g}*sqrt({h})} }{\\var{h-j^2}}\\text{,} \\]</p>\n<p style=\"text-align: left;\">Das k&ouml;nnen wir noch vereinfachen zu</p>\n<p style=\"text-align: center;\">\\[ \\var{o}\\sqrt{\\var{h}}-{\\var{l}}\\text{.} \\]</p>\n<p></p>\n<h4>e)</h4>\n<p>Wir rechnen</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{t}-\\sqrt{\\var{u}}}{\\var{t}+\\sqrt{\\var{u}}}$ =</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{t}-\\sqrt{\\var{u}}}{\\var{t}+\\sqrt{\\var{u}}}\\times\\frac{\\var{t}-\\sqrt{\\var{u}}}{\\var{t}-\\sqrt{\\var{u}}}=\\frac{(\\var{t}-\\sqrt{\\var{u}})(\\var{t}-\\sqrt{\\var{u}})}{(\\var{t}+\\sqrt{\\var{u}})(\\var{t}-\\sqrt{\\var{u}})}=\\frac{\\var{t^2}-\\var{2t}\\sqrt{\\var{u}}+\\var{u}}{\\var{t^2}-\\var{u}}$.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{t^2}-\\var{2t}\\sqrt{\\var{u}}+\\var{u}}{\\var{t^2}-\\var{u}}=\\frac{\\var{t^2+u}-\\var{2t}{\\sqrt{\\var{u}}}}{\\var{t^2-u}}$.</p>\n<div data-jme-visible=\"{gcd_frac} &gt; 1\">\n<p style=\"text-align: left;\">K&uuml;rzen mit ${\\var{gcd_frac}}$ ergibt dann</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{num_simp_1}-\\simplify{{num_simp_2}*sqrt({u})}}{\\var{denom_simp}}$.&nbsp;</p>\n</div>\n<h4>f)</h4>\n<p style=\"text-align: left;\">Um die Br&uuml;che $\\displaystyle\\frac{\\var{p}}{\\sqrt{\\var{p}}+\\sqrt{\\var{p^3}}}+\\frac{\\var{r_coprime}}{\\var{s_coprime}}$ zu addieren, m&uuml;ssen wir sie zun&auml;chst auf einen gemeinsamen Nenner bringen.</p>\n<p style=\"text-align: left;\">Beachten Sie, dass $\\sqrt{\\var{p^3}}$ umgeschrieben werden kann als $\\var{p}\\sqrt{\\var{p}}$, das bedeutet, dass wir den ersten Bruch schreiben k&ouml;nnen als</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{p}}{\\sqrt{\\var{p}}+\\var{p}\\sqrt{\\var{p}}} \\text{.} \\]</p>\n<p style=\"text-align: left;\">Das vereinfachen wir weiter zu</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{p}}{\\var{p+1}\\sqrt{\\var{p}}} \\text{.} \\]</p>\n<p>Deshalb haben wir als Zwischenergebnis $\\displaystyle\\frac{\\var{p}}{{\\var{p+1}}\\sqrt{\\var{p}}}+\\frac{\\var{r_coprime}}{\\var{s_coprime}}$.</p>\n<p>Um die Addition durchzuf&uuml;hren, m&uuml;ssen wir den zweiten Bruch auf denselben Nenner bringen. Wir erweitern mit $\\simplify{{p_p1}sqrt({p})}$, also</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{r_coprime}}{\\var{s_coprime}}\\cdot\\simplify[all,!simplifyFractions,!sqrtDivision]{{p_p1}*sqrt({p})/({p_p1}*sqrt({p}))}=\\frac{\\simplify{{one}*sqrt({p})}}{\\var{two}\\sqrt{\\var{p}}}$,</p>\n<p>Jetzt erhalten wir</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{p}}{\\var{p+1}\\sqrt{\\var{p}}}+\\frac{\\simplify{{one}*sqrt({p})}}{\\var{two}\\sqrt{\\var{p}}}= \\frac{\\var{p}+\\simplify{{one}*sqrt({p})}}{\\var{p+1}\\sqrt{\\var{p}}} \\]</p>\n<p>Weil $\\displaystyle\\sqrt{a}\\cdot\\sqrt{a}=a$ ist, k&ouml;nnen wir $\\sqrt{\\var{p}}$ im Z&auml;hler ausklammern:</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\sqrt{\\var{p}}(\\sqrt{\\var{p}}+\\var{r_coprime*p_p1})}{\\var{p+1}\\sqrt{\\var{p}}}\\text{.} \\]</p>\n<p>Wir k&uuml;rzen jetzt mit $\\sqrt{\\var{p}}$ und bekommen</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\sqrt{\\var{p}}+\\var{one}}{\\var{p+1}} \\text{.} \\]</p>", "rulesets": {}, "extensions": [], "variables": {"r": {"name": "r", "group": "Part f", "definition": "random(1..2)", "description": "", "templateType": "anything"}, "t": {"name": "t", "group": "Part e", "definition": "random(1..6)", "description": "", "templateType": "anything"}, "g": {"name": "g", "group": "Part d", "definition": "k*(h-j^2)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Part b", "definition": "random(2,4,5)", "description": "<p>Random number.</p>", "templateType": "anything"}, "d": {"name": "d", "group": "Part c", "definition": "random(2..10)", "description": "", "templateType": "anything"}, "one": {"name": "one", "group": "Part f", "definition": "r_coprime*p_p1", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Part b", "definition": "repeat(random(7..11 except 8 except 9 except 10),5)", "description": "<p>Random numbers, not square.</p>", "templateType": "anything"}, "r_coprime": {"name": "r_coprime", "group": "Part f", "definition": "r/gcd(r,s)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Part b", "definition": "random(2,3,4)", "description": "<p>Random number.</p>", "templateType": "anything"}, "p": {"name": "p", "group": "Part f", "definition": "random(2..5 except 4)", "description": "", "templateType": "anything"}, "s": {"name": "s", "group": "Part f", "definition": "p+1", "description": "", "templateType": "anything"}, "h": {"name": "h", "group": "Part d", "definition": "j^2+random(-3..3 except 0)", "description": "", "templateType": "anything"}, "two": {"name": "two", "group": "Part f", "definition": "s_coprime*p_p1", "description": "", "templateType": "anything"}, "num_simp_1": {"name": "num_simp_1", "group": "Part e", "definition": "(t^2+u)/gcd_frac", "description": "<p>Simplified numerical term of numerator.</p>", "templateType": "anything"}, "c_coprime": {"name": "c_coprime", "group": "Part c", "definition": "c/gcd_cdf", "description": "", "templateType": "anything"}, "k": {"name": "k", "group": "Part d", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "p_p1": {"name": "p_p1", "group": "Part f", "definition": "ceil(p/(s_coprime))", "description": "", "templateType": "anything"}, "gcd_num": {"name": "gcd_num", "group": "Part e", "definition": "gcd(t^2+u, 2*t)", "description": "<p>GCD of terms in numerator.</p>", "templateType": "anything"}, "o": {"name": "o", "group": "Part d", "definition": "g/(h-j^2)", "description": "", "templateType": "anything"}, "ghi": {"name": "ghi", "group": "Part e", "definition": "t^2-u", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Part b", "definition": "repeat(random(3..5 except 4),5)", "description": "<p>Random numbers, not square.</p>", "templateType": "anything"}, "l": {"name": "l", "group": "Part d", "definition": "(g*j)/(h-j^2)", "description": "", "templateType": "anything"}, "j": {"name": "j", "group": "Part d", "definition": "random(1..5)", "description": "", "templateType": "anything"}, "square_nums": {"name": "square_nums", "group": "Part a", "definition": "shuffle([4,9,16,25])", "description": "<p>List of square numbers.</p>", "templateType": "anything"}, "u": {"name": "u", "group": "Part e", "definition": "random(2..10 except 4 except 9) ", "description": "", "templateType": "anything"}, "denom_simp": {"name": "denom_simp", "group": "Part e", "definition": "ghi/gcd_frac", "description": "<p>Simplified denominator.</p>", "templateType": "anything"}, "c": {"name": "c", "group": "Part c", "definition": "random(1..10)", "description": "", "templateType": "anything"}, "qwerty": {"name": "qwerty", "group": "Part f", "definition": "(r_coprime*p_p1)-1", "description": "", "templateType": "anything"}, "gcd_cdf": {"name": "gcd_cdf", "group": "Part c", "definition": "gcd(c,df)", "description": "", "templateType": "anything"}, "df": {"name": "df", "group": "Part c", "definition": "d*f", "description": "", "templateType": "anything"}, "m_lcm": {"name": "m_lcm", "group": "Part b", "definition": "b[0]*lcm(n,m)", "description": "<p>Multiple of LCM of n amd m.</p>", "templateType": "anything"}, "prime_nums": {"name": "prime_nums", "group": "Part a", "definition": "shuffle([3,5,7,11,13])", "description": "<p>List of prime numbers.</p>", "templateType": "anything"}, "n_lcm": {"name": "n_lcm", "group": "Part b", "definition": "a[0]*lcm(n,m)", "description": "<p>Multiple of LCM of n amd m.</p>", "templateType": "anything"}, "s_coprime": {"name": "s_coprime", "group": "Part f", "definition": "s/gcd(r,s)", "description": "", "templateType": "anything"}, "df_coprime": {"name": "df_coprime", "group": "Part c", "definition": "df/gcd_cdf", "description": "", "templateType": "anything"}, "gcd_frac": {"name": "gcd_frac", "group": "Part e", "definition": "gcd(gcd_num, t^2-u)", "description": "<p>GCD of all terms in fraction.</p>", "templateType": "anything"}, "num_simp_2": {"name": "num_simp_2", "group": "Part e", "definition": "2t/gcd_frac", "description": "<p>Simplified surd term of numerator.</p>", "templateType": "anything"}, "f": {"name": "f", "group": "Part c", "definition": "random(2..10 except 4 except 9)", "description": "", "templateType": "anything"}, "num": {"name": "num", "group": "Part b", "definition": "m_lcm*n_lcm", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": [], "variable_groups": [{"name": "Part a", "variables": ["square_nums", "prime_nums"]}, {"name": "Part b", "variables": ["n", "m", "n_lcm", "m_lcm", "a", "b", "num"]}, {"name": "Part c", "variables": ["c", "d", "f", "df", "gcd_cdf", "c_coprime", "df_coprime"]}, {"name": "Part d", "variables": ["g", "h", "j", "l", "k", "o"]}, {"name": "Part f", "variables": ["p", "r", "s", "r_coprime", "s_coprime", "p_p1", "qwerty", "one", "two"]}, {"name": "Part e", "variables": ["t", "u", "ghi", "denom_simp", "gcd_num", "gcd_frac", "num_simp_1", "num_simp_2"]}], "functions": {}, "preamble": {"js": "", "css": ".fractiontable table {\n  width: 20%; \n  padding: 0px; \n  border-width: 0px; \n  layout: fixed;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  width: 15%; \n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n.fractiontable .tdeq \n{\n  width: 5%; \n  border-bottom: 0px;\n  font-size: x-large;\n}\n\n\n.fractiontable th {\n    background-color:#aaa;\n}\n/*Fix the height of all cells EXCEPT table-headers to 40px*/\n.fractiontable td {\n    height:40px;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Vereinfachen Sie die folgenden Ausdr&uuml;cke und schreiben Sie sie als ganze Zahlen.</p>\n<p><em>i)</em></p>\n<p></p>\n<p>$\\displaystyle\\frac{\\sqrt{\\simplify{{square_nums[0]}{prime_nums[0]}}}}{\\sqrt{\\var{prime_nums[0]}}}=$&nbsp;[[0]]</p>\n<p></p>\n<p><em>ii)</em></p>\n<p>$\\displaystyle\\frac{\\sqrt{\\simplify{{square_nums[1]}{prime_nums[1]}}}}{\\sqrt{\\var{prime_nums[1]}}}=$&nbsp;[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "sqrt{square_nums[0]}", "maxValue": "sqrt{square_nums[0]}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["si-fr", "eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "sqrt{square_nums[1]}", "maxValue": "sqrt{square_nums[1]}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["si-fr", "eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Vereinfachen Sie den Ausdruck, indem Sie ihn mit einer ganzen Zahl als Nenner schreiben.</p>\n<p></p>\n<div class=\"fractiontable\">\n<table style=\"text-align: right; width: 348.417px;\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td rowspan=\"2\" style=\"width: 111px;\">$\\displaystyle\\frac{\\sqrt{\\var{n_lcm}}}{\\sqrt{\\var{m_lcm}}}=$</td>\n<td class=\"tdnum\" style=\"text-align: center; width: 190.417px;\">$\\surd$[[0]]</td>\n</tr>\n<tr>\n<td class=\"tddenom\" style=\"text-align: center; width: 190.417px;\">[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n</div>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a[0]*b[0]", "maxValue": "a[0]*b[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b[0]", "maxValue": "b[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Schreiben Sie den Nenner als ganze Zahl und k&uuml;rzen Sie so weit wie m&ouml;glich.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{c}}{\\var{d}\\sqrt{\\var{f}}} = \\frac{a\\sqrt{\\var{f}}}{b}\\text{,} \\]</p>\n<p style=\"text-align: left;\">wobei</p>\n<p style=\"text-align: left;\">$a =$&nbsp;[[0]]</p>\n<p style=\"text-align: left;\">$b =$&nbsp;[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c_coprime", "maxValue": "c_coprime", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "df_coprime", "maxValue": "df_coprime", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Schreiben Sie den Ausdruck in der Form $a\\sqrt{b}+m$, mit ganzen Zahlen $a$, $b$ und $c$.</p>\n<p></p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{g}}{(\\sqrt{\\var{h}}+\\var{j})}$ =&nbsp;</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$\\displaystyle\\var{o}\\sqrt{\\var{h}}+\\var{l}$</p>", "<p>$\\displaystyle-\\var{l}-\\var{o}\\sqrt{\\var{h}}$</p>", "<p>$\\displaystyle\\var{o}\\sqrt{\\var{h}}-\\var{l}$</p>", "<p>$\\displaystyle\\var{l}-\\sqrt{\\var{h}}$</p>"], "matrix": [0, "0", "2", 0], "distractors": ["", "", "", ""]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Schreiben Sie mit einer ganzen Zahl als Nenner und vereinfachen Sie so weit wie m&ouml;glich:</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{t}-\\sqrt{\\var{u}}}{\\var{t}+\\sqrt{\\var{u}}} = \\frac{a-b\\sqrt{\\var{u}}}{c} \\text{,} \\]</p>\n<p style=\"text-align: left;\">wobei:</p>\n<p style=\"text-align: left;\">$a = $&nbsp;[[0]]</p>\n<p style=\"text-align: left;\">$b = $&nbsp;[[1]]</p>\n<p style=\"text-align: left;\">$c = $ [[2]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "num_simp_1", "maxValue": "num_simp_1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "num_simp_2", "maxValue": "num_simp_2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "denom_simp", "maxValue": "denom_simp", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Vervollst&auml;ndigen Sie diese Addition, indem Sie zuerst den ersten Summanden als Bruch mit einer ganzen Zahl im Nenner ausdr&uuml;cken. Die Antwort soll ein Bruch mit einer ganzen Zahl im Nenner sein.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{p}}{\\sqrt{\\var{p}}+\\sqrt{\\var{p^3}}}+\\frac{\\var{r_coprime}}{\\var{s_coprime}} \\]</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$\\displaystyle\\frac{\\sqrt{\\var{p}}+\\var{r_coprime*p_p1}}{\\var{p+1}}$</p>", "<p>$\\displaystyle\\frac{\\sqrt{\\var{p}}(\\sqrt{\\var{p}}+\\var{r_coprime*p_p1})}{\\var{p+1}\\sqrt{\\var{p}}}$</p>", "<p>$\\displaystyle\\frac{\\var{p}+\\var{qwerty}\\sqrt{\\var{p}}}{\\var{p}\\sqrt{\\var{p}}}$</p>", "<p>$\\displaystyle\\frac{\\sqrt{\\var{p}}+\\var{r_coprime*p_p1}}{\\var{p}}$</p>"], "matrix": ["3", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}