// Numbas version: finer_feedback_settings
{"name": "Indices: combinations of rules (non-algebraic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: combinations of rules (non-algebraic)", "tags": ["exponent", "exponents", "fractional", "index", "index laws", "Index Laws", "indices", "power", "powers"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Simplify the following without the use of a calculator. Write your answer in index form using ^ to signify powers.</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "primes[0]", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "primes[2]", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "primes[1]", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..12 except [root,p,n])", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..12 except [root,p])", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2..12 except root)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "random([root,p,n,m,a,b,c])", "description": "", "templateType": "anything", "can_override": false}, "primes": {"name": "primes", "group": "Ungrouped variables", "definition": "shuffle([2,3,5,7,11,13,17,19])[0..3]", "description": "", "templateType": "anything", "can_override": false}, "root": {"name": "root", "group": "Ungrouped variables", "definition": "random([2,3,4,5])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["primes", "a", "b", "c", "root", "p", "n", "m", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "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>If you would like a challenge, then here is a question for you. Use your index laws rather than your calculator to see how this simplifies:</p>\n<p>$\\displaystyle \\frac{\\sqrt[\\var{root}]{\\var{a}^\\var{n}\\times\\var{b}\\times\\var{a}^\\var{root-n}}}{\\var{c}^\\var{-m}\\times\\var{a}\\times\\var{b}^0\\times\\var{b}^{1/\\var{root}}}\\times \\left(\\frac{\\var{a}^\\var{p}\\times\\var{b}}{\\var{c}}\\right)^\\var{m}+\\var{ans}(\\var{a}^\\var{root-n}\\times\\var{b}^\\var{m+root}\\times\\var{c}^\\var{n+m})^0-\\frac{\\var{a}^\\var{p*m-n}\\times\\var{a}^\\var{n}}{\\var{b}^\\var{-m}}$ =&nbsp;[[0]]</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "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>The order shown is simply one order in which to do things, you also don't need to use as many lines as are shown here. There are only three terms in this expression. We will simplify them individually and then combine them.</p>\n<p>The middle term $\\var{ans}(\\var{a}^\\var{root-n}\\times\\var{b}^\\var{m+root}\\times\\var{c}^\\var{n+m})^0$ has a power of zero acting on a bracket so this can be reduced to simply $\\var{ans}\\times 1=\\var{ans}$.</p>\n<p>The third term is slightly more complicated in that we will need to add indices (because we are multiplying expressions with the same base) and deal with negative indices:</p>\n<p>\\begin{align}&amp;\\frac{\\var{a}^\\var{p*m-n}\\times\\var{a}^\\var{n}}{\\var{b}^\\var{-m}}\\\\<br/>&amp;=\\frac{\\var{a}^\\color{red}{\\var{p*m}}}{\\var{b}^\\var{-m}}\\\\<br/>&amp;=\\var{a}^\\var{p*m}\\times\\color{red}{\\var{b}^\\var{m}}\\end{align}</p>\n<p>The first term is more complicated and we will use all of our index laws to simplify it:<br/>\\begin{align}&amp;\\frac{\\sqrt[\\var{root}]{\\var{a}^\\var{n}\\times\\var{b}\\times\\var{a}^\\var{root-n}}}{\\var{c}^\\var{-m}\\times\\var{a}\\times\\var{b}^0\\times\\var{b}^{1/\\var{root}}}\\times \\left(\\frac{\\var{a}^\\var{p}\\times\\var{b}}{\\var{c}}\\right)^\\var{m}\\\\<br/>&amp;=\\frac{\\left(\\var{a}^\\var{n}\\times\\var{b}\\times\\var{a}^\\var{root-n}\\right)^{\\color{red}{1/\\var{root}}}}{\\var{c}^\\var{-m}\\times\\var{a}\\times \\color{red}{1}\\times\\var{b}^{1/\\var{root}}}\\times \\left(\\frac{\\var{a}^\\color{red}{\\var{p*m}}\\times\\var{b}^\\color{red}{\\var{m}}}{\\var{c}^\\color{red}{\\var{m}}}\\right)\\\\<br/>&amp;=\\frac{\\left(\\var{a}^\\color{red}{\\var{root}}\\times\\var{b}\\right)^{1/\\var{root}}\\color{red}{\\times}\\var{a}^\\var{p*m}\\times\\var{b}^\\var{m}}{\\var{c}^\\var{-m}\\times\\var{a}\\times\\var{b}^{1/\\var{root}}\\color{red}{\\times} \\var{c}^\\var{m}}\\\\<br/>&amp;=\\frac{\\var{a}^\\color{red}{1}\\times\\var{b}^\\color{red}{1/\\var{root}}\\color{red}{\\times}\\var{a}^\\var{p*m}\\times\\var{b}^\\var{m}}{\\var{c}^\\color{red}{\\var{0}}\\times\\var{a}\\times\\var{b}^{1/\\var{root}}}\\\\<br/>&amp;=\\frac{\\var{a}^\\color{red}{\\var{1+p*m}}\\times\\var{b}^\\color{red}{\\var{m}+1/\\var{root}}}{\\color{red}{1}\\times\\var{a}\\times\\var{b}^{1/\\var{root}}}\\\\<br/>&amp;=\\var{a}^\\color{red}{\\var{p*m}}\\times\\var{b}^\\color{red}{\\var{m}}\\\\<br/>\\end{align}</p>\n<p>This is the same as the third term.</p>\n<p>So far we have reduced our original expression to $\\var{a}^\\color{red}{\\var{p*m}}\\times\\var{b}^\\color{red}{\\var{m}}+\\var{ans}-\\var{a}^\\color{red}{\\var{p*m}}\\times\\var{b}^\\color{red}{\\var{m}}$ but this is simply $\\var{ans}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans}", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "maxlength": {"length": "5", "partialCredit": 0, "message": "<p>Your answer is&nbsp;longer than necessary.</p>"}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "resources": []}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}