// Numbas version: exam_results_page_options
{"name": "Numbers VI: surds (rationalising the denominator - simple)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/radianintroduction_a_Cn82Zth.png", "/srv/numbas/media/question-resources/radianintroduction_a_Cn82Zth.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a1", "a2", "a3", "b1", "b2", "b3", "c1", "c2", "d1", "d2", "d3", "d4", "e1", "e2", "e3", "e4", "e5", "f1", "f2", "f3", "f4", "f5", "f6", "f7", "f8"], "name": "Numbers VI: surds (rationalising the denominator - simple)", "tags": [], "preamble": {"css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}", "js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');"}, "advice": "<p>The best approach here is to first rationalise the denominator. We do this by multiplying both top and bottom by an approriate value.</p>\n<p>For example to rationalise the denominator for an expression like $\\frac{a}{\\sqrt{b}}$, we multiply numerator and denominator by $\\sqrt{b}$ to get $\\frac{a\\sqrt{b}}{b}$</p>\n<p>Similarly to rationalise the denominator for an expression like $\\frac{a+\\sqrt{b}}{c+\\sqrt{d}}$, we multiply numerator and denominator by $c-\\sqrt{d}$ to get $\\frac{(a+\\sqrt{b})(c-\\sqrt{d})}{c^2-d}$</p>\n<p>Dont forget that:</p>\n<p>$\\sqrt{a \\times b} = \\sqrt{a} \\times \\sqrt{b}$</p>\n<p>$\\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}}$</p>\n<p>$(a+\\sqrt{b})(a-\\sqrt{b})=a^2-b$</p>\n<p>Remember to check that your answer is in its simplest form.</p>\n<p>If you need to simplify a large surd, try to see if it has a square number as a factor. Say $c=a^2\\times b$. Then we have that $\\sqrt{c} = \\sqrt{a^2b} = \\sqrt{a^2}\\sqrt{b} = a\\sqrt{b}$.</p>", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>$\\frac{\\var{a1}}{\\sqrt{\\var{a2}}}=$<fraction><numerator>[[0]]$\\sqrt{}$[[1]]</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>Firstly, multipliy both the denominator and the numerator by the surd in the denominator as follows:</p>\n<p>$\\frac{\\var{a1}}{\\sqrt{\\var{a2}}}=\\frac{\\var{a1}}{\\sqrt{\\var{a2}}}\\times\\frac{\\sqrt{\\var{a2}}}{\\sqrt{\\var{a2}}}$</p>\n<p>By doing so the original fraction doesn't change in value. This is because the invented multiplier of $\\frac{\\sqrt{\\var{a2}}}{\\sqrt{\\var{a2}}}$ simplifies to equal $1$, and $\\frac{\\var{a1}}{\\sqrt{\\var{a2}}}\\times1=\\frac{\\var{a1}}{\\sqrt{\\var{a2}}}$.</p>\n<p>The result of the first procedure is:</p>\n<p>$\\frac{\\var{a1}\\sqrt{\\var{a2}}}{\\var{a2}}$</p>\n<p>This leaves a rational number in the denominator and transfers the irrational surd into the numerator, hence the process name of 'rationalising the denominator'.</p>\n<p>The final answer, therefore, is:</p>\n<p>$\\frac{\\simplify{{a1}/{a3}}\\sqrt{\\var{a2}}}{\\simplify{{a2}/{a3}}}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "a1/a3", "minValue": "a1/a3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "a2", "minValue": "a2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "a2/a3", "minValue": "a2/a3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "<p></p>\n<p>$\\frac{\\var{b1}}{\\sqrt{\\var{b2}}}=$<fraction><numerator>[[0]]$\\sqrt{}$[[1]]</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>As before, multipliy both the denominator and the numerator by the surd in the denominator as follows:</p>\n<p>$\\frac{\\var{b1}}{\\sqrt{\\var{b2}}}=\\frac{\\var{b1}}{\\sqrt{\\var{b2}}}\\times\\frac{\\sqrt{\\var{b2}}}{\\sqrt{\\var{b2}}}$</p>\n<p>By doing so the original fraction doesn't change in value. This is because the invented multiplier of $\\frac{\\sqrt{\\var{b2}}}{\\sqrt{\\var{b2}}}$ simplifies to equal $1$, and $\\frac{\\var{b1}}{\\sqrt{\\var{b2}}}\\times1=\\frac{\\var{b1}}{\\sqrt{\\var{b2}}}$.</p>\n<p>The result of the first procedure is:</p>\n<p>$\\frac{\\var{b1}\\sqrt{\\var{b2}}}{\\var{b2}}$</p>\n<p>This leaves a rational number in the denominator and transfers the irrational surd into the numerator, hence the process name of 'rationalising the denominator'.</p>\n<p>The final answer, therefore, is:</p>\n<p>$\\frac{\\simplify{{b1}/{b3}}\\sqrt{\\var{b2}}}{\\simplify{{b2}/{b3}}}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "b1/b3", "minValue": "b1/b3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "b2", "minValue": "b2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "b2/b3", "minValue": "b2/b3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "<p>$\\frac{\\var{c1}}{\\sqrt{\\var{c2}}}=$<fraction><numerator>[[0]]$\\sqrt{}$[[1]]</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>As before, multipliy both the denominator and the numerator by the surd in the denominator as follows:</p>\n<p>$\\frac{\\var{c1}}{\\sqrt{\\var{c2}}}=\\frac{\\var{c1}}{\\sqrt{\\var{c2}}}\\times\\frac{\\sqrt{\\var{c2}}}{\\sqrt{\\var{c2}}}$</p>\n<p>By doing so the original fraction doesn't change in value. This is because the invented multiplier of $\\frac{\\sqrt{\\var{c2}}}{\\sqrt{\\var{c2}}}$ simplifies to equal $1$, and $\\frac{\\var{c1}}{\\sqrt{\\var{c2}}}\\times1=\\frac{\\var{c1}}{\\sqrt{\\var{c2}}}$.</p>\n<p>The result of the first procedure is:</p>\n<p>$\\frac{\\var{c1}\\sqrt{\\var{c2}}}{\\var{c2}}$</p>\n<p>This leaves a rational number in the denominator and transfers the irrational surd into the numerator, hence the process name of 'rationalising the denominator'.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{c1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{c2}", "minValue": "{c2}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "c2", "minValue": "c2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "<p>$\\frac{\\var{d4}\\sqrt{\\var{d1}}}{\\sqrt{\\var{d2}}}=$<fraction><numerator>[[0]]$\\sqrt{}$[[1]]</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>As before, multipliy both the denominator and the numerator by the surd in the denominator as follows:</p>\n<p>$\\frac{\\var{d4}\\sqrt{\\var{d1}}}{\\sqrt{\\var{d2}}}=\\frac{\\var{d4}\\sqrt{\\var{d1}}}{\\sqrt{\\var{d2}}}\\times\\frac{\\sqrt{\\var{d2}}}{\\sqrt{\\var{d2}}}$</p>\n<p>By doing so the original fraction doesn't change in value. This is because the invented multiplier of $\\frac{\\sqrt{\\var{d2}}}{\\sqrt{\\var{d2}}}$ simplifies to equal $1$, and $\\frac{\\var{d4}\\sqrt{\\var{d1}}}{\\sqrt{\\var{d2}}}\\times1=\\frac{\\var{d4}\\sqrt{\\var{d1}}}{\\sqrt{\\var{d2}}}$.</p>\n<p>The result of the first procedure is:</p>\n<p>$\\frac{\\var{d4}\\sqrt{\\var{d1}\\times\\var{d2}}}{\\var{d2}}$</p>\n<p>This leaves a rational number in the denominator and transfers the irrational surd into the numerator, hence the process name of 'rationalising the denominator'.</p>\n<p>The final answer, therefore, is:</p>\n<p>$\\frac{\\simplify{{d4}/{d3}}\\sqrt{\\simplify{{d1}*{d2}}}}{\\simplify{{d2}/{d3}}}$ which simplifies to ${\\simplify{{d4}/{d3}}\\sqrt{\\simplify{{d1}*{d2}}}}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "d4/d3", "minValue": "d4/d3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "d1*d2", "minValue": "d1*d2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "d2/d3", "minValue": "d2/d3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "<p>$\\frac{\\var{e1}\\sqrt{\\var{e2}}}{\\var{e3}\\sqrt{\\var{e4}}}=$<fraction><numerator>[[0]]$\\sqrt{}$[[1]]</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>As before, multipliy both the denominator and the numerator by the surd in the denominator as follows:</p>\n<p>$\\frac{\\var{e1}\\sqrt{\\var{e2}}}{\\var{e3}\\sqrt{\\var{e4}}}=\\frac{\\var{e1}\\sqrt{\\var{e2}}}{\\var{e3}\\sqrt{\\var{e4}}}\\times\\frac{\\sqrt{\\var{e4}}}{\\sqrt{\\var{e4}}}$</p>\n<p>By doing so the original fraction doesn't change in value. This is because the invented multiplier of $\\frac{\\sqrt{\\var{e4}}}{\\sqrt{\\var{e4}}}$ simplifies to equal $1$, and $\\frac{\\var{e1}\\sqrt{\\var{e2}}}{\\var{e3}\\sqrt{\\var{e4}}}\\times1=\\frac{\\var{e1}\\sqrt{\\var{e2}}}{\\var{e3}\\sqrt{\\var{e4}}}$.</p>\n<p>The result of the first procedure is:</p>\n<p>$\\frac{\\var{e1}\\sqrt{\\var{e2}\\times\\var{e4}}}{\\var{e3}\\times\\var{e4}}$</p>\n<p>This leaves a rational number in the denominator and transfers the irrational surd into the numerator, hence the process name of 'rationalising the denominator'.</p>\n<p>The final answer, therefore, is:</p>\n<p>$\\frac{\\simplify{{e1}/{e5}}\\sqrt{\\simplify{{e2}*{e4}}}}{\\simplify{{e3}*{{e4}/{e5}}}}$, <span data-jme-visible=\"{e3}*({e4}/{e5})=1 and ({e1}/{e5})&gt;1\">which simplifies to&nbsp;${\\simplify{{e1}/{e5}}\\sqrt{\\simplify{{e2}*{e4}}}}$.</span>&nbsp;<span data-jme-visible=\"{e3}*({e4}/{e5})=1 and ({e1/e5})=1\">which simplifies to $\\sqrt{{\\simplify{{e2}*{e4}}}}$.</span>&nbsp;<span data-jme-visible=\"{e3}*({e4}/{e5})&gt;1 and ({e1}/{e5})=1\">which simplifies to $\\frac{\\sqrt{\\simplify{{e2}*{e4}}}}{\\simplify{{e3}*{{e4}/{e5}}}}$.</span></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "e1/e5", "minValue": "e1/e5", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "e2*e4", "minValue": "e2*e4", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "e3*e4/e5", "minValue": "e3*e4/e5", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "<p>$\\frac{\\var{f1}\\sqrt{\\var{f6}}}{\\var{f2}\\sqrt{\\var{f5}}}=$<fraction><numerator>[[0]]$\\sqrt{}$[[1]]</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>As before, multipliy both the denominator and the numerator by the surd in the denominator as follows:</p>\n<p>$\\frac{\\var{f1}\\sqrt{\\var{f6}}}{\\var{f2}\\sqrt{\\var{f5}}}=\\frac{\\var{f1}\\sqrt{\\var{f6}}}{\\var{f2}\\sqrt{\\var{f5}}}\\times\\frac{\\sqrt{\\var{f5}}}{\\sqrt{\\var{f5}}}$</p>\n<p>By doing so the original fraction doesn't change in value. This is because the invented multiplier of $\\frac{\\sqrt{\\var{f5}}}{\\sqrt{\\var{f5}}}$ simplifies to equal $1$, and $\\frac{\\var{f1}\\sqrt{\\var{f6}}}{\\var{f2}\\sqrt{\\var{f5}}}\\times1=\\frac{\\var{f1}\\sqrt{\\var{f6}}}{\\var{f2}\\sqrt{\\var{f5}}}$.</p>\n<p>The result of the first procedure is:</p>\n<p>$\\frac{\\var{f1}\\sqrt{\\var{f6}\\times\\var{f5}}}{\\var{f2}\\times\\var{f5}}$</p>\n<p>This leaves a rational number in the denominator and transfers the irrational surd into the numerator, hence the process name of 'rationalising the denominator'.</p>\n<p>The result is further simplified to:</p>\n<p>$\\frac{\\var{f1}\\sqrt{\\var{f1}\\times\\var{f1}\\times\\var{f2}\\times\\var{f2}\\times\\var{f3}\\times\\var{f4}\\times\\var{f7}}}{\\var{f2}\\times\\var{f3}\\times\\var{f1}\\times\\var{f1}}=\\frac{\\var{f1}\\times\\var{f1}\\times\\var{f2}\\sqrt{\\var{f3}\\times\\var{f4}\\times\\var{f7}}}{\\var{f2}\\times\\var{f3}\\times\\var{f1}\\times\\var{f1}}=\\frac{\\sqrt{\\var{f7}}\\times\\sqrt{\\var{f3}\\times\\var{f4}}}{\\var{f3}}$</p>\n<p></p>\n<p>The final answer, therefore, is:</p>\n<p>$\\frac{\\simplify{{f7}^0.5}\\sqrt{\\simplify{{f3}*{f4}}}}{\\var{f3}}$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "(f7)^0.5", "minValue": "(f7)^0.5", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "(f4*f3)", "minValue": "(f4*f3)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "f3", "minValue": "f3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p><strong>'Rationalise the denominators' of the following expressions and present&nbsp;each in the form $\\frac{a\\sqrt{b}}{c}$ where a, b and c are integers,&nbsp;where a and c have no common factors.</strong></p>\n<p>With simpler operations, it's easier to manipulate fractions which contain surds in the numerator instead of the denominator. The process of transferring the surd from the denominator to the numerator is called 'ratinoalising the denominator'. If you've never done a question like this before, click on 'Show steps' and follow the solutions, or click 'Reveal answers' at the bottom of the page and study the Advice section. Once you are confident with the method, click 'Try another question like this one' and complete the questions without clicking for help.</p>\n<p><em><strong>Note:</strong><strong></strong> the final answer may simplify in such a way that the number $1$ is in the place of&nbsp;numerator coefficient, in the denominator or both. In these cases the $1$ wouldn't be written in the final answer but this question requires that you write it into the given gaps.</em></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"f1": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "f2": {"definition": "random(2..5 except f1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "f3": {"definition": "random([2,3,5,7,11,13] except f2 except f1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f3", "description": ""}, "f4": {"definition": "random([2,3,5,7] except f1 except f2 except f3)", "templateType": "anything", "group": "Ungrouped variables", "name": "f4", "description": ""}, "f5": {"definition": "f3*f1*f1", "templateType": "anything", "group": "Ungrouped variables", "name": "f5", "description": ""}, "f6": {"definition": "f8*f2*f2", "templateType": "anything", "group": "Ungrouped variables", "name": "f6", "description": ""}, "f7": {"definition": "random([4,9,16] except f3)", "templateType": "anything", "group": "Ungrouped variables", "name": "f7", "description": ""}, "f8": {"definition": "f4*f7", "templateType": "anything", "group": "Ungrouped variables", "name": "f8", "description": ""}, "b1": {"definition": "random(2..19 except b2)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random(2,3,5,6,7,10,11,13,14,15,17,19,21,23,22,26,29)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "b3": {"definition": "gcd(b2,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "d4": {"definition": "random([4,6,10,12,14,20,22,26] except [2*d1])", "templateType": "anything", "group": "Ungrouped variables", "name": "d4", "description": ""}, "d2": {"definition": "d4/2", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "d3": {"definition": "gcd(d4,d2)", "templateType": "anything", "group": "Ungrouped variables", "name": "d3", "description": ""}, "d1": {"definition": "random([2,3,5,7,11,13,17,19,23,29] except [c1])", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "a1": {"definition": "random(2..19 except a2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a3": {"definition": "gcd(a1,a2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "a2": {"definition": "random(2,3,5,6,7,10,11,13,14,15,17,19,21,23,22,26,29)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "e5": {"definition": "gcd(e1,e3*e4)", "templateType": "anything", "group": "Ungrouped variables", "name": "e5", "description": ""}, "e4": {"definition": "random([2,3,5,7] except [e2])", "templateType": "anything", "group": "Ungrouped variables", "name": "e4", "description": ""}, "c1": {"definition": "random(2,3,5,7,11,13,17,19,23,29)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "e1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "e1", "description": ""}, "e3": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "e3", "description": ""}, "e2": {"definition": "random([2,3,5,7])", "templateType": "anything", "group": "Ungrouped variables", "name": "e2", "description": ""}, "c2": {"definition": "random([2,3,5,7,11,13,17,19,23,29] except [c1])", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}}, "metadata": {"description": "<p>Practice rationalising a single-term, surd denominator.</p>\n<p><em>Adapted from&nbsp;</em><strong>'Surds&nbsp;2'</strong><em> by&nbsp;</em><strong>Joshua Boddy</strong>.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}]}]}], "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}]}