// Numbas version: exam_results_page_options {"name": " Algebra IV: Properties of indices (1) - Multiplication/Division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "h", "j"], "name": " Algebra IV: Properties of indices (1) - Multiplication/Division", "tags": [], "preamble": {"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}", "js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');"}, "advice": "

Recall the laws of indices to help solve the problems:

\n

$x^a \\times x^b = x^{a+b}$

\n

$x^a \\div x^b = x^{a-b}$

\n

$x^{-a} = \\frac{1}{x^a}$

\n

$(x^a)^b = x^{ab}$

\n

$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$

\n

$x^\\frac{a}{b} = (\\sqrt[b]{x})^{a}$

\n

$x^0 = 1$

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

$x^\\var{a} \\times x^\\var{b}$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use the following indices law to help answer this question:

\n

$x^a \\times x^b = x^{a+b}$

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "answer": "x^({a}+{b})", "marks": 1, "showCorrectAnswer": true, "checkingtype": "absdiff", "type": "jme", "checkvariablenames": false}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

$p^\\var{c} \\times p^\\var{d}$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use the following law to help answer this question:

\n

$x^a \\times x^b = x^{a+b}$

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all", "scripts": {}, "answer": "p^({c}+{d})", "marks": 1, "showCorrectAnswer": true, "checkingtype": "absdiff", "type": "jme", "checkvariablenames": false}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

$(\\var{a}k^\\var{b})^\\var{f}$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use the following law to answer this question:

\n

$(ax^b)^c = a^cx^{bc}$

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all", "scripts": {}, "answer": "{a}^{f}*k^({b}*{f})", "marks": 1, "showCorrectAnswer": true, "checkingtype": "absdiff", "type": "jme", "checkvariablenames": false}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

$y^{1/\\var{a}} \\times y^{1/\\var{b}}$

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use the following law:

\n

$x^a \\times x^b = x^{a+b}$

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all", "scripts": {}, "answer": "y^(({a}+{b})/({a}*{b}))", "marks": 1, "showCorrectAnswer": true, "checkingtype": "absdiff", "type": "jme", "checkvariablenames": false}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

$c^\\var{a}$$c^\\var{b}$

\n

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use the following law:

\n

$x^a \\div x^b = x^{a-b}$

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all", "scripts": {}, "answer": "c^({a}-{b})", "marks": 1, "showCorrectAnswer": true, "checkingtype": "absdiff", "type": "jme", "checkvariablenames": false}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

$\\var{a}h^\\var{c}$$\\var{b}h^\\var{d}$

\n

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use the following law to answer this question:

\n

$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all", "scripts": {}, "answer": "{a}/{b}*h^({c}-{d})", "marks": 1, "showCorrectAnswer": true, "checkingtype": "absdiff", "type": "jme", "checkvariablenames": false}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

$(4d)^\\var{g}$$(2d)^\\var{h}$

\n

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

This question differs from part f due to the brackets. Using principles of BODMAS, the brackets need to be expanded first. 

\n

$(4d)^\\var{g}$ expands to $4^\\var{g}d^\\var{g}$ and $(2d)^\\var{h}$ expands to $2^\\var{h}d^\\var{h}$

\n

Now you are left with a simple division question as follows:

\n

$\\frac{4^{\\var{g}}d^{\\var{g}}}{2^{\\var{h}}d^{\\var{h}}}$

\n

\n

Use the principle:

\n

$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$ to answer the question.

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all", "scripts": {}, "answer": "(4^{g})/(2^{h})*d^{g-h}", "marks": 1, "showCorrectAnswer": true, "checkingtype": "absdiff", "type": "jme", "checkvariablenames": false}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

$(6p^{-\\var{f}})^{\\var{g}}$$(9p^{-\\var{j}})^{\\var{h}}$

\n

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Using principles of BODMAS, the brackets need to be expanded first. 

\n

$(6p)^{-\\var{f}}$ expands to $6^{-\\var{f}}p^{-\\var{g}}$ and $(9p)^{-\\var{j}}$ expands to $9^{-\\var{j}}p^{-\\var{j}}$

\n

Now you are left with a simple division question as follows:

\n

$\\frac{6^{-\\var{f}}p^{-\\var{g}}}{9^{-\\var{j}}p^{-\\var{j}}}$

\n

\n

Use the principle:

\n

$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$ to answer the question.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all", "scripts": {}, "answer": "(6^{g})/(9^{h})*p^{h*j-g*f}", "marks": 1, "showCorrectAnswer": true, "checkingtype": "absdiff", "type": "jme", "checkvariablenames": false}], "extensions": [], "statement": "

Simplify each of the following expressions, giving your answer in its simplest form.

\n

Recall your knowledge of indices laws.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(0..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..9 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-9..-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(2..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "h": {"definition": "random(2..5 except g)", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}, "j": {"definition": "random(2..3 except f)", "templateType": "anything", "group": "Ungrouped variables", "name": "j", "description": ""}}, "metadata": {"description": "

Simplifying indices

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Nasir Firoz Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/909/"}]}]}], "contributors": [{"name": "Nasir Firoz Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/909/"}]}