// Numbas version: exam_results_page_options
{"name": "Jinhua's copy of Indices 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "statement": "<p>Simplify each of these, giving your answer&nbsp;in its simplest form.</p>", "parts": [{"showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "marks": 1, "answer": "x^({a}+{b})", "showpreview": true, "prompt": "<p>$x^\\var{a} \\times x^\\var{b}$</p>", "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "marks": 1, "answer": "p^({c}+{d})", "showpreview": true, "prompt": "<p>$p^\\var{c} \\times p^\\var{d}$</p>", "answersimplification": "all", "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "marks": 1, "answer": "{a}^{f}*k^({b}*{f})", "showpreview": true, "prompt": "<p>$(\\var{a}k^\\var{b})^\\var{f}$</p>", "answersimplification": "all", "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "marks": 1, "answer": "y^(({a}+{b})/({a}*{b}))", "showpreview": true, "prompt": "<p>$y^{\\simplify{1/{a}}} \\times y^{\\simplify{1/{b}}}$</p>", "answersimplification": "all", "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "marks": 1, "answer": "c^({a}-{b})", "showpreview": true, "prompt": "<p><fraction><numerator>$c^\\var{a}$</numerator><denominator>$c^\\var{b}$</denominator></fraction></p>\n<p></p>", "answersimplification": "all", "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "marks": 1, "answer": "{a}/{b}*h^({c}-{d})", "showpreview": true, "prompt": "<p><fraction><numerator>$\\var{a}h^\\var{c}$</numerator><denominator>$\\var{b}h^\\var{d}$</denominator></fraction></p>\n<p></p>", "answersimplification": "all", "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "marks": 1, "answer": "(4^{g})/(2^{h})*d^{g-h}", "showpreview": true, "prompt": "<p><fraction><numerator>$(4d)^\\var{g}$</numerator><denominator>$(2d)^\\var{h}$</denominator></fraction></p>\n<p></p>", "answersimplification": "all", "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "variableReplacements": [], "scripts": {}, "vsetrange": [0, 1], "marks": 1, "answer": "(6^{g})/(9^{h})*p^{h*j-g*f}", "showpreview": true, "prompt": "<p><fraction><numerator>$(6p^{-\\var{f}})^{\\var{g}}$</numerator><denominator>$(9p^{-\\var{j}})^{\\var{h}}$</denominator></fraction></p>\n<p></p>", "answersimplification": "all", "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff"}], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "variables": {"a": {"definition": "random(2..9)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "a"}, "b": {"definition": "random(2..9 except a)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "b"}, "f": {"definition": "random(2..3)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "f"}, "h": {"definition": "random(2..5 except g)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "h"}, "j": {"definition": "random(2..3 except f)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "j"}, "d": {"definition": "random(-9..-1)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "d"}, "g": {"definition": "random(2..5)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "g"}, "c": {"definition": "random(0..9)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "c"}}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "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}"}, "rulesets": {}, "metadata": {"description": "<p>Simplifying indices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "question_groups": [{"questions": [], "pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered"}], "name": "Jinhua's copy of Indices 3", "tags": [], "advice": "<p><em><strong>Recall the laws of indices to help solve the problems:</strong></em></p>\n<p>$x^a \\times x^b = x^{a+b}$</p>\n<p>$x^a \\div x^b = x^{a-b}$</p>\n<p>$x^{-a} = \\frac{1}{x^a}$</p>\n<p>$(x^a)^b = x^{ab}$</p>\n<p>$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$</p>\n<p>$x^\\frac{a}{b} = \\sqrt[b]{x^a}$</p>\n<p>$x^0 = 1$</p>\n<p></p>\n<p><em><strong>Worked Solutions:</strong></em></p>\n<p>Part a)<em></em></p>\n<p>$x^\\var{a} \\times x^\\var{b}=x^{(\\var{a}+\\var{b})}=x^{\\simplify{{a}+{b}}}$</p>\n<p>Part b)</p>\n<p>$p^\\var{c} \\times p^\\var{d}=p^{(\\var{c}+\\var{d})}=p^{\\simplify{{c}+{d}}}$</p>\n<p>Part c)</p>\n<p>For this question, recall that the exterior power is applied to both the coefficient and the variable within brackets.</p>\n<p>$(\\var{a}k^\\var{b})^\\var{f}=(\\var{a}^\\var{f}k^{\\var{b}\\times\\var{f}})=\\simplify{{a}^{f}}k^{\\simplify{{b}*{f}}}$</p>\n<p>Part d)</p>\n<p>$y^{\\simplify{1/{a}}} \\times y^{\\simplify{1/{b}}}=y^{\\frac{1}{\\var{a}}+\\frac{1}{\\var{b}}}=y^{\\simplify{{{a}+{b}}/{{a}*{b}}}}$</p>\n<p>Part e)</p>\n<p>$\\frac{c^{\\var{a}}}{c^{\\var{b}}}=c^{\\var{a}-\\var{b}}=c^{\\simplify{{a}-{b}}}$</p>\n<p>Part f)</p>\n<p>$\\frac{\\var{a}h^{\\var{c}}}{\\var{b}h^{\\var{d}}}=\\frac{\\var{a}}{\\var{b}}\\times h^{\\var{c}-\\var{d}}=\\simplify{{a}/{b}}h^{\\simplify{{c}-{d}}}$</p>\n<p>Part g)</p>\n<p>$\\frac{({4d})^{\\var{g}}}{({2d})^{\\var{h}}}=\\frac{4^{\\var{g}}{d}^{\\var{g}}}{2^{\\var{h}}{d}^{\\var{h}}}=\\frac{\\simplify{4^{{g}}}{d}^{\\var{g}}}{\\simplify{2^{{h}}}{d}^{\\var{h}}}=\\simplify{{4^{g}}/{2^{h}}}d^{\\var{g}-\\var{h}}=\\simplify{{4^{g}}/{2^{h}}}d^{\\simplify{{g}-{h}}}$</p>\n<p>Part h)</p>\n<p>$\\frac{({6p^{-\\var{f}}})^{\\var{g}}}{({9p^{-\\var{j}}})^{\\var{h}}}=\\frac{6^{\\var{g}}{p}^{-\\var{f}\\times\\var{g}}}{9^{\\var{h}}{p}^{-\\var{j}\\times\\var{h}}}=\\left(\\frac{\\simplify{6^{g}}}{\\simplify{9^{h}}}\\right)\\times \\frac{p^{\\simplify{-{f}*{g}}}}{p^{\\simplify{-{j}*{h}}}}=\\simplify{{6^{g}}/{9^{h}}}\\times p^{\\simplify{-{f}*{g}}-\\simplify{-{j}*{h}}}=\\simplify{{6^{g}}/{9^{h}}}p^{\\simplify{{-{f}*{g}}-{-{j}*{h}}}}$</p>\n<p>&nbsp;</p>", "showQuestionGroupNames": false, "extensions": [], "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "h", "j"], "functions": {}, "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}]}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}