// Numbas version: finer_feedback_settings
{"name": "Multiplication and division of numbers in standard index form", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Multiplication and division of numbers in standard index form", "tags": ["division", "indices", "multiplication", "standard form", "standard index form", "taxonomy"], "metadata": {"description": "<p>Multiply two numbers in standard form, then divide two numbers in standard form.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Simplify,&nbsp;leaving the result in&nbsp;standard index form.</p>", "advice": "<p>We use the rules for multiplying and dividing powers.</p>\n<p style=\"padding-left: 30px;\">To <strong>multiply</strong> powers we <strong>add</strong>,&nbsp;for example&nbsp;$10^2 \\times 10^6 = 10^{(2+6)} = 10^{8}$.</p>\n<p style=\"padding-left: 30px;\">To&nbsp;<strong>divide</strong> powers we <strong>subtract</strong>, for example $10^2 \\div 10^6 = 10^{(2-6)} =&nbsp;10^{-4}$.</p>\n<p style=\"padding-left: 30px;\">&nbsp;</p>\n<h4>a)</h4>\n<p>\\[ \\begin{align} (\\var{int} \\times&nbsp;10^\\var{ran - 3}&nbsp;) \\times (\\var{int - 11}&nbsp;\\times 10^\\var{ran})&nbsp; &amp;= \\var{int} \\times (\\var{int-11}) \\times 10^{(\\var{ran - 3} + \\var{ran})} \\\\&amp;= (\\var{int*(int-11)}) \\times 10^{\\var{ran - 3 + ran}} \\end{align} \\]</p>\n<p>We want to write this number in standard form so we move decimal place one place to the left to get</p>\n<p>\\[(\\var{int*(int-11)/10}) \\times 10^{(\\var{ran - 3 + ran} + 1)} = (\\var{int*(int-11)/10}) \\times 10^{\\var{ran -&nbsp;2 + ran}} \\text{.}\\]</p>\n<h4>&nbsp;</h4>\n<h4>b)</h4>\n<p>Similarly,</p>\n<p>\\[ \\begin{align} (\\var{int*int} \\times&nbsp;10^\\var{ran -4}&nbsp;) \\div (\\var{int}&nbsp;\\times 10^\\var{ran - 2})&nbsp; &amp;= \\var{int*int} \\div \\var{int} \\times 10^{(\\var{ran - 4} - \\var{ran -2})} \\\\&amp;= \\var{int} \\times 10^{-2} \\text{.}\\end{align} \\]</p>", "rulesets": {}, "extensions": [], "variables": {"int": {"name": "int", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "ran": {"name": "ran", "group": "Ungrouped variables", "definition": "random([6,7,8,9,10])", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ran", "int"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p style=\"padding-left: 30px;\">$(\\var{int} \\times&nbsp;10^\\var{ran - 3}&nbsp;) \\times (\\var{int - 11}&nbsp;\\times 10^\\var{ran})&nbsp; =$&nbsp; [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{(int)*(int - 11)/10}*10^{{2*ran - 2}}", "answerSimplification": "!collectnumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "musthave": {"strings": ["*10^"], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["10^3", "10^4", "10^5", "10^6", "10^7", "10^(3)", "10^(4)", "10^(5)", "10^(6)", "10^(7)", "81", "24", "28", "30"], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p style=\"padding-left: 30px;\">$(\\var{int*int} \\times&nbsp;10^\\var{ran -4}&nbsp;) \\div (\\var{int}&nbsp;\\times 10^\\var{ran - 2})&nbsp; =$&nbsp; [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{int}*10^(-2)", "answerSimplification": "!collectnumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "musthave": {"strings": ["*10^"], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["/", "10^4", "10^(4)", "10^5", "10^(5)", "10^6", "10^(6)"], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "sortAnswers": false}], "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}]}