// Numbas version: exam_results_page_options
{"name": "Indices: fractional powers (non-algebraic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "name": "Indices: fractional powers (non-algebraic)", "functions": {}, "showQuestionGroupNames": false, "variables": {"zeros": {"templateType": "anything", "group": "Ungrouped variables", "name": "zeros", "definition": "random(2..5)", "description": ""}, "onezeros": {"templateType": "anything", "group": "Ungrouped variables", "name": "onezeros", "definition": "10^zeros", "description": ""}, "base1": {"templateType": "anything", "group": "Ungrouped variables", "name": "base1", "definition": "random(2,3,5,7,11,13,17,19)", "description": ""}, "square": {"templateType": "anything", "group": "Ungrouped variables", "name": "square", "definition": "random([[4,2],[9,3],[16,4],[25,5],[36,6],[49,7],[64,8],[81,9],[121,11],[144,12]])", "description": ""}, "root3": {"templateType": "anything", "group": "Ungrouped variables", "name": "root3", "definition": "random(2..12)", "description": ""}, "root2": {"templateType": "anything", "group": "Ungrouped variables", "name": "root2", "definition": "random(2..12)", "description": ""}, "cube": {"templateType": "anything", "group": "Ungrouped variables", "name": "cube", "definition": "random([[8,4],[27,9]])", "description": ""}, "power2": {"templateType": "anything", "group": "Ungrouped variables", "name": "power2", "definition": "random(2..12 except root2)", "description": ""}, "base2": {"templateType": "anything", "group": "Ungrouped variables", "name": "base2", "definition": "random([2,3,5,7] except base1)", "description": ""}, "power3": {"templateType": "anything", "group": "Ungrouped variables", "name": "power3", "definition": "random(2..12 except [root2,root3])", "description": ""}, "primes": {"templateType": "anything", "group": "Ungrouped variables", "name": "primes", "definition": "shuffle([11,13,17,19])[0..2] ", "description": ""}, "base4": {"templateType": "anything", "group": "Ungrouped variables", "name": "base4", "definition": "primes[1]", "description": ""}, "base3": {"templateType": "anything", "group": "Ungrouped variables", "name": "base3", "definition": "primes[0]", "description": ""}, "root1": {"templateType": "anything", "group": "Ungrouped variables", "name": "root1", "definition": "random(4..12)", "description": ""}}, "variable_groups": [], "type": "question", "parts": [{"showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "vsetrangepoints": 5, "notallowed": {"strings": ["sqrt", "root"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "marks": 1, "showCorrectAnswer": true, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\nvar rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\nif(!m) {\n  this.multCredit(0,'Your answer is not in the form $x^y$.');\n}\n"}}, "checkvariablenames": false, "type": "jme", "showpreview": false, "musthave": {"strings": ["^"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "answer": "{base1}^(1/2)", "variableReplacements": [], "answersimplification": "basic", "maxlength": {"message": "<p>Your answer is&nbsp;longer than necessary.</p>", "length": "11", "partialCredit": 0}, "checkingaccuracy": "0.00001", "expectedvariablenames": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "steps": [{"variableReplacements": [], "prompt": "<p>Given \\[(\\sqrt{\\var{base1}})^2=\\var{base1}=(\\var{base1}^{1/2})^2\\]</p>\n<p>we can say \\[\\sqrt{\\var{base1}}=\\var{base1}^{1/2}\\]</p>\n<p>Which we would type in as $\\var{base1}\\wedge(1/2)$.</p>", "showCorrectAnswer": true, "scripts": {}, "type": "information", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "prompt": "<p>By using the definition of the square root you should see that $(\\sqrt{\\var{base1}})^2=\\var{base1}$.</p>\n<p>By using index laws you should see that $(\\var{base1}^{1/2})^2=\\var{base1}$.</p>\n<p>The above equations imply that $\\sqrt{\\var{base1}}$ can also be written as &nbsp;[[0]].</p>\n<p></p>\n<p>Note: If you want to use a fraction as a power you should use brackets to surround your power, for example, type 12^(2/3) for $12^\\frac{2}{3}$.</p>", "scripts": {}, "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "vsetrangepoints": 5, "notallowed": {"strings": ["^0", "sqrt", "root"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "marks": 1, "showCorrectAnswer": true, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\nvar rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\nif(!m) {\n  this.multCredit(0,'Your answer is not in the form $x^y$.');\n}\n"}}, "checkvariablenames": false, "type": "jme", "showpreview": true, "musthave": {"strings": ["^"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "answer": "{base2}^(1/3)", "variableReplacements": [], "answersimplification": "basic", "maxlength": {"message": "<p>Your answer is&nbsp;longer than necessary.</p>", "length": "11", "partialCredit": 0}, "checkingaccuracy": "0.00001", "expectedvariablenames": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "steps": [{"variableReplacements": [], "prompt": "<p>Given \\[(\\sqrt[3]{\\var{base2}})^3=\\var{base2}=(\\var{base2}^{1/3})^3\\]</p>\n<p>we can say \\[\\sqrt[3]{\\var{base2}}=\\var{base2}^{1/3}\\]</p>\n<p>Which we would type in as $\\var{base2}\\wedge(1/3)$.</p>", "showCorrectAnswer": true, "scripts": {}, "type": "information", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "prompt": "<p>By using the definition of the cube root you should see that $(\\sqrt[3]{\\var{base2}})^3=\\var{base2}$.</p>\n<p>By using index laws you should see that $(\\var{base2}^{1/3})^3=\\var{base2}$.</p>\n<p>The above equations imply that $\\sqrt[3]{\\var{base2}}$ can also be written as &nbsp;[[0]].</p>\n<p></p>\n<p>Note: If you want to use a fraction as a power you should use brackets to surround your power, for example, type 12^(2/3) for $12^\\frac{2}{3}$.</p>", "scripts": {}, "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "vsetrangepoints": 5, "notallowed": {"strings": ["^0", "sqrt", "root"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "marks": 1, "showCorrectAnswer": true, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\nvar rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\nif(!m) {\n  this.multCredit(0,'Your answer is not in the form $x^y$.');\n}\n"}}, "checkvariablenames": false, "type": "jme", "showpreview": true, "musthave": {"strings": ["^"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "answer": "{base3}^(1/{root1})", "variableReplacements": [], "answersimplification": "basic", "maxlength": {"message": "<p>Your answer is&nbsp;longer than necessary.</p>", "length": "11", "partialCredit": 0}, "checkingaccuracy": "0.00001", "expectedvariablenames": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "steps": [{"variableReplacements": [], "prompt": "<p>By the same reasoning as used in the above questions we have $\\sqrt[n]{a}=a^{\\frac{1}{n}}$.&nbsp;</p>", "showCorrectAnswer": true, "scripts": {}, "type": "information", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "prompt": "<p>Use the same approach you used in the above questions to simplify the following in index form.</p>\n<p><br/>$\\sqrt[\\var{root1}]{\\var{base3}}$ =&nbsp;[[0]]</p>\n<p></p>", "scripts": {}, "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "vsetrangepoints": 5, "notallowed": {"strings": ["^0", "sqrt", "root"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "marks": 1, "showCorrectAnswer": true, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\nvar rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\nif(!m) {\n  this.multCredit(0,'Your answer is not in the form $x^y$.');\n}\n"}}, "checkvariablenames": false, "type": "jme", "showpreview": true, "musthave": {"strings": ["^"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "answer": "{base4}^({power2}/{root2})", "variableReplacements": [], "answersimplification": "basic", "maxlength": {"message": "<p>Your answer is&nbsp;longer than necessary.</p>", "length": "11", "partialCredit": 0}, "checkingaccuracy": "0.00001", "expectedvariablenames": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "steps": [{"variableReplacements": [], "prompt": "<p><em>Convert the root to a fractional power and then use the index laws to deal with the two different powers.</em></p>\n<hr/>\n<p>For example, \\[\\sqrt[3]{2}^5=(2^{\\frac{1}{3}})^5=2^{\\frac{5}{3}}\\]&nbsp;</p>", "showCorrectAnswer": true, "scripts": {}, "type": "information", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "prompt": "<p>$\\displaystyle\\left(\\sqrt[\\var{root2}]{\\var{base4}}\\right)^\\var{power2}$ =&nbsp;[[0]]</p>\n<p></p>", "scripts": {}, "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "gaps": [{"checkingtype": "absdiff", "vsetrangepoints": 5, "notallowed": {"strings": ["^0", "sqrt", "root"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "marks": 1, "showCorrectAnswer": true, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\nvar rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\nif(!m) {\n  this.multCredit(0,'Your answer is not in the form $x^y$.');\n}\n"}}, "checkvariablenames": false, "type": "jme", "showpreview": true, "musthave": {"strings": ["^"], "message": "<p>Use ^ for powers. Input your answer in index form.</p>", "partialCredit": 0, "showStrings": false}, "answer": "{base1}^({power3}/{root3})", "variableReplacements": [], "answersimplification": "basic", "maxlength": {"message": "<p>Your answer is&nbsp;longer than necessary.</p>", "length": "11", "partialCredit": 0}, "checkingaccuracy": "0.00001", "expectedvariablenames": [], "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "steps": [{"variableReplacements": [], "prompt": "<p><em>Convert the root to a fractional power and then use the index laws to deal with the two different powers.</em></p>\n<hr/>\n<p>For example, \\[\\sqrt[3]{2^5}=(2^5)^{\\frac{1}{3}}=2^{\\frac{5}{3}}\\]&nbsp;</p>", "showCorrectAnswer": true, "scripts": {}, "type": "information", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "prompt": "<p>$\\sqrt[\\var{root3}]{\\var{base1}^\\var{power3}}$ =&nbsp;[[0]]</p>\n<p></p>", "scripts": {}, "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "minValue": "square[1]", "variableReplacements": [], "maxValue": "square[1]", "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "marks": 1, "type": "numberentry", "allowFractions": false}, {"correctAnswerFraction": false, "minValue": "cube[1]", "variableReplacements": [], "maxValue": "cube[1]", "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "marks": 1, "type": "numberentry", "allowFractions": false}, {"correctAnswerFraction": false, "minValue": "10", "variableReplacements": [], "maxValue": "10", "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "marks": 1, "type": "numberentry", "allowFractions": false}], "variableReplacements": [], "steps": [{"variableReplacements": [], "prompt": "<p><em>Convert the denominator of the fractional power to a root.&nbsp;</em></p>\n<hr/>\n<p>For example, $27^{\\frac{2}{3}}=\\sqrt[3]{27}^2=3^2=9$.</p>", "showCorrectAnswer": true, "scripts": {}, "type": "information", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "prompt": "<p>By interpreting the denominator of the fractional power as an n<sup>th</sup> root, determine the value of the following:</p>\n<p><br/>$\\var{square[0]}^{\\frac{1}{2}}$ =&nbsp;[[0]]</p>\n<p>$\\var{cube[0]}^{\\frac{2}{3}}$ =&nbsp;[[1]]</p>\n<p>$\\var{onezeros}^{\\frac{1}{\\var{zeros}}}$ =&nbsp;[[2]]</p>", "scripts": {}, "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst"}], "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0}], "statement": "<p>Simplify the following without the use of a calculator. Write your answer in index form using ^ to signify powers.</p>", "ungrouped_variables": ["primes", "base1", "base2", "base3", "base4", "root1", "root2", "power2", "root3", "power3", "square", "cube", "zeros", "onezeros"], "rulesets": {}, "advice": "", "tags": ["exponent", "exponents", "fractional", "index", "index laws", "indices", "power", "powers", "rational", "roots"], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "", "notes": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "contributors": [{"name": "Marlon Arcila", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/321/"}]}]}], "contributors": [{"name": "Marlon Arcila", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/321/"}]}