// Numbas version: exam_results_page_options
{"name": "Indices: negative powers (non-algebraic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: negative powers (non-algebraic)", "tags": ["exponent", "exponents", "index", "index laws", "Index Laws", "indices", "negative", "power", "powers"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Use ^ to signify powers.</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"base1": {"name": "base1", "group": "Ungrouped variables", "definition": "primes[0]", "description": "", "templateType": "anything", "can_override": false}, "base2": {"name": "base2", "group": "Ungrouped variables", "definition": "primes[1]", "description": "", "templateType": "anything", "can_override": false}, "base3": {"name": "base3", "group": "Ungrouped variables", "definition": "primes[2]", "description": "", "templateType": "anything", "can_override": false}, "base": {"name": "base", "group": "Ungrouped variables", "definition": "random(2,3,5,10)", "description": "", "templateType": "anything", "can_override": false}, "primes": {"name": "primes", "group": "Ungrouped variables", "definition": "shuffle([2,3,5,7,11,13,17])[0..3] ", "description": "", "templateType": "anything", "can_override": false}, "power3": {"name": "power3", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "power2": {"name": "power2", "group": "Ungrouped variables", "definition": "random(-6..-2)", "description": "", "templateType": "anything", "can_override": false}, "power1": {"name": "power1", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "<p>2..6</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["primes", "base1", "power1", "base2", "power2", "base3", "power3", "base"], "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>For an insight into negative indices, consider the following table:&nbsp;</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\">index form</td>\n<td style=\"text-align: center;\">$\\var{base}^3$</td>\n<td style=\"text-align: center;\">$\\var{base}^2$</td>\n<td style=\"text-align: center;\">$\\var{base}^1$</td>\n<td style=\"text-align: center;\">$\\var{base}^0$</td>\n<td style=\"text-align: center;\">$\\var{base}^{-1}$</td>\n<td style=\"text-align: center;\">$\\var{base}^{-2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">result</td>\n<td style=\"text-align: center;\">$\\var{base^3}$</td>\n<td style=\"text-align: center;\">$\\var{base^2}$</td>\n<td style=\"text-align: center;\">$\\var{base}$</td>\n<td style=\"text-align: center;\">$1$</td>\n<td style=\"text-align: center;\">[[0]]</td>\n<td style=\"text-align: center;\">[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p>Notice each time the power decreases by $1$, the result is divided by $\\var{base}$. Using this idea, and / for division, fill in the rest of the table.</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>Each time you reduce the power by 1 you divide the result by the base, that is $\\var{base}$. Following this pattern:<br/><br/>\\[\\var{base}^{-1}=\\frac{1}{\\var{base}}\\]</p>\n<p>and&nbsp;</p>\n<p>\\[\\var{base}^{-2}=\\frac{1}{\\var{base}}\\div\\var{base}=\\frac{1}{\\var{base}^2}\\]</p>\n<p></p>\n<hr/>\n<p></p>\n<p>Note you could input the second expression as $1/\\var{base}/\\var{base}$ or $1/\\var{base}\\wedge2$ or $1/\\var{base^2}$.</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": "1/{base}", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.0000001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["^0", "^1", "^-"], "showStrings": false, "partialCredit": 0, "message": "<p>Don't use negative powers here in this table.</p>"}, "valuegenerators": []}, {"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": "1/{base}^2", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.0000001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["^0", "^1", "^-"], "showStrings": false, "partialCredit": 0, "message": "<p>Don't use negative powers here in this table.</p>"}, "valuegenerators": []}], "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>The&nbsp;fraction $\\displaystyle\\frac{1}{\\var{base1}^\\var{power1}}$ can be written using a negative index as&nbsp;[[0]].</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>Notice:<br/>\\[1=\\var{base1}^0=\\var{base1}^{\\var{power1}-\\var{power1}}=\\var{base1}^{\\var{power1}} \\var{base1}^{-\\var{power1}}\\]</p>\n<p>&nbsp;That is, we have</p>\n<p>\\[1=\\var{base1}^{\\var{power1}} \\var{base1}^{-\\var{power1}}\\]</p>\n<p>by dividing both sides of this equation by $\\var{base1}^{\\var{power1}}$ we get</p>\n<p>\\[\\frac{1}{\\var{base1}^{\\var{power1}}}=\\var{base1}^{-\\var{power1}}\\]</p>\n<hr/>\n<p>In general, we have $\\displaystyle\\frac{1}{a^c}=a^{-c}$.</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": "{base1}^{-power1}", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.0000001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "notallowed": {"strings": ["^0", "/", "^1"], "showStrings": false, "partialCredit": 0, "message": "<p>Write your answer with a negative index.</p>"}, "mustmatchpattern": {"pattern": "?^?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}], "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>The expression $\\var{base2}^{\\var{power2}}$ can be written without a negative index as the fraction &nbsp;[[0]].</p>\n<p></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>Notice:<br/>\\[1=\\var{base2}^0=\\var{base2}^{\\var{power2}+\\var{-power2}}=\\var{base2}^{\\var{power2}} \\var{base2}^{\\var{-power2}}\\]</p>\n<p>&nbsp;That is, we have</p>\n<p>\\[1=\\var{base2}^{\\var{power2}} \\var{base2}^{\\var{-power2}}\\]</p>\n<p>by dividing both sides of this equation by $\\var{base2}^{\\var{-power2}}$ we get</p>\n<p>\\[\\frac{1}{\\var{base2}^{\\var{-power2}}}=\\var{base2}^{\\var{power2}}\\]</p>\n<hr/>\n<p>In general, we have $\\displaystyle\\frac{1}{a^c}=a^{-c}$.</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": "1/{base2}^{-power2}", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.0000001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["/"], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["^0", "^1", "^-"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "valuegenerators": []}], "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>The expression $\\displaystyle\\frac{1}{\\var{base3}^{\\var{-power3}}}$ can be written without a negative index and without the use of a fraction.</p>\n<p>The simplest way to write&nbsp;$\\displaystyle\\frac{1}{\\var{base3}^{\\var{-power3}}}$ in index form would be&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>You can think of the negative power as forcing the term to the other part of the fraction (if it was on top it goes to the bottom and if it was on the bottom it goes to the top) except now it has a&nbsp;positive power.</p>\n<hr/>\n<p>We can see this by using our rules for dividing fractions:</p>\n<p>\\[\\frac{1}{\\var{base3}^\\var{-power3}}=\\frac{1}{\\left(\\frac{1}{\\var{base3}^\\var{power3}}\\right)}=1\\div{\\left(\\frac{1}{\\var{base3}^\\var{power3}}\\right)}=1\\times \\left(\\frac{{\\var{base3}^\\var{power3}}}{1}\\right)=\\var{base3}^\\var{power3}\\]</p>\n<hr/>\n<p>In general, we have $\\displaystyle\\frac{1}{a^c}=a^{-c}$ and $\\displaystyle\\frac{1}{a^{-c}}=a^{c}$.</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": "{base3}^{power3}", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "<p>Use index form.</p>"}, "notallowed": {"strings": ["/", "^-", "^0", "^1"], "showStrings": false, "partialCredit": 0, "message": "<p>Don't use fractions or negative indices.</p>"}, "mustmatchpattern": {"pattern": "?^?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "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/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}