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\\var{x1} ^\\var{n} &nbsp;$</p>\n\t\t\t", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "basic", "scripts": {}, "answer": "{x1^n}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "\n\t\t\t<p>Evaluate</p>\n\t\t\t      <p>$ (\\simplify{ &nbsp; {x2}/{x3} &nbsp; })^\\var{n3} $</p>\n\t\t\t      <p>&nbsp;</p>\n\t\t\t", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{x2/x3}^{n3}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"prompt": "\n\t\t\t<p>Express each of the following in the form &nbsp;$ &nbsp;\\var{x4}^n &nbsp;$.</p>\n\t\t\t      <p>For negative values of \"n\" enclose your answer in brackets &nbsp;&nbsp;</p>\n\t\t\t      <p>i) &nbsp; $\\var{x4}^{\\var{n1}} \\times &nbsp;\\var{x4}^{\\var{n2}} $ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; [[0]]</p>\n\t\t\t      <p>ii) &nbsp; $\\var{x4}^{\\var{n4}} \\div &nbsp;\\var{x4}^{\\var{n5}} $ &nbsp; &nbsp; &nbsp; [[1]]</p>\n\t\t\t      <p>iii) &nbsp;&nbsp;$ (\\var{x4}^{\\var{n6}})^\\var{n7} &nbsp;$ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; [[2]]&nbsp;</p>\n\t\t\t      <p>&nbsp;</p>\n\t\t\t", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "basic", "scripts": {}, "answer": "{x4}^{n1+n2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", 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"parts": [{"vsetrangepoints": 5, "prompt": "<p>What is $x^\\var{a}.x^\\var{b}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "x^{c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "<p>Give the algebraic expression for the following question:</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "a+b", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}}, "metadata": {"description": "<p>Just multiply 2 algebraic expressions with indices</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Julie's copy of Index Multiplication Mixed Variables", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "a1", "c", "b", "q", "p", "r", "b1", "c1"], "tags": ["rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "<p>What is : $x^\\var{a}.y^\\var{b}.z^\\var{c}.x^\\var{a1}.z^\\var{c1}.y^\\var{b1}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [1, 1.1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "x^{p}*y^{q}*z^{r}", "marks": "3", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "<p>Give the algebraic answer to the following expression:</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "q": {"definition": "b+b1", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "description": ""}, "c": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "a1": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "p": {"definition": "a+a1", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "r": {"definition": "c+c1", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "b1": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "c1": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}}, "metadata": {"description": "<p>This question mixes up three variables in multiplication.Note the checking range is between 1 and 1.1 in the Accuracy and String restrictions section of the Part. 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If the checking range is left as between 0 and 1 then large powers will be very small and be less than the accuracy setting.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "BCU Headstart Algebra Q5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Peter Samuels", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/91/"}], "tags": [], "progress": "in-progress", "metadata": {"notes": "", "description": "<p>Substituting a variable into a formula</p>"}, "statement": "<p><span>Find the value of b when a = 2, using the formula b = (6 - a)/2.</span></p>", "advice": "<p>This is a diagnostic test so there is no advice.</p>", 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"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({c}x+{a})/{a}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Note: to input the answer \"$x=y+2$\" the \"$x=$\" is already given and you just need to input \"$y+2$\".</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..9 except[a,b])", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..9 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "atimesc": {"definition": "{a}*{c}", "templateType": "anything", "group": "Ungrouped variables", "name": "atimesc", "description": ""}}, "metadata": {"description": "<p>Transposition</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "transposing formula 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d"], "tags": ["REBEL", "rebel", "Rebel", "rebelmaths", "transpose"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepsPenalty": 1, "prompt": "<p>$\\simplify[std]{{a}y + {b}x = {c} + {d}xy}\\;$</p>\n<p>$y =$ [[0]].</p>\n<p>You can click on \"Show steps\" for more information, but you will lose one mark if you do so.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>To re-arrange $ay + bx = c + dxy$ we should first collect all of the terms involving $y$ to the one side</p>\n          <p>$ay - dxy = c - bx$</p>\n          <p>we should then factorize out $y$ to find</p>\n          <p>$y(a-dx) = c - bx$</p>\n          <p>and then divide by $a-dx$ to get $y$ on its own</p>\n          <p>$y = \\frac{c - bx}{a - dx}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({c} - {b}x)/({a} - {d}x)", "marks": "5", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Rearrange the following equation to make $y$ the subject.&nbsp;</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-10..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-10..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-10..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"description": "<p>Another transposition question.</p>\n<p>rebalmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Julie's copy of Algebra: Expansion of two brackets (one linear)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "a1", "b1", "c1"], "tags": ["algebra", "Algebra", "algebraic manipulation", "expansion of brackets", "expansion of the product of two linear terms", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "\n  <p>1. Using the method given by Show steps we have:</p>\n  <p>\\[\\simplify[std]{ {a}x*({c}x+{d})}=\\simplify[std]{{a*c}x^2+{a*d}x}\\]</p>\n  <p>2.</p>\n  <p>\\[\\simplify[std]{ ({a1}x+{b1})*({c1}x)}=\\simplify[std]{{a1*c1}x^2+{b1*c1}x}\\]</p>\n  <p>&nbsp;</p>\n  <p>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n      <p>$\\simplify[std]{({a}x)({c}x+{d})}=\\;$[[0]].</p>\n      <p>$\\simplify[std]{({a1}x+{b1})({c1}x)}=\\;$[[1]].</p>\n      <p>Your answers should be quadratics in $x$ and should not include any brackets.</p>\n      <p>You can click on Show steps to get more information, but you will lose one mark if you do so.</p>\n      ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>\\[ax(cx+d)=acx^2+adx\\]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"notallowed": {"message": "<p>Do not include brackets in your answer. Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.</p>", "showStrings": false, "strings": ["("], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a*c}x^2+{a*d}x", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "maxlength": {"length": 13, "message": "<p>Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.</p>", "partialCredit": 0}}, {"notallowed": {"message": "<p>Do not include brackets in your answer. Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.</p>", "showStrings": false, "strings": ["("], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a1*c1}*x^2+{b1*c1}*x", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "maxlength": {"length": 13, "message": "<p>Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.</p>", "partialCredit": 0}}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Expand the following to give quadratics in $x$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-9..9 except [0,c])", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "a1": {"definition": "random(-5..5 except [0,a])", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "b1": {"definition": "random(-9..9 except [0,c])", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "c1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}}, "metadata": {"description": "<p>Expansion of two brackets</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Expansion of two brackets: Linear 2 positive coefficients", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d"], "tags": ["algebra", "algebraic manipulation", "expansion of brackets", "expansion of the product of two linear terms", "Rebel", "REBEL", "rebel", "rebelmaths"], "advice": "\n  <p>Using the method given by Show steps we have:</p>\n  <p>\\[\\begin{eqnarray*}\\simplify[std]{ ({a}x+{b})({c}x+{d})}&amp;=&amp;\\simplify[std]{{a}x*({c}x+{d})+{b}({c}x+{d})}\\\\&amp;=&amp;\\simplify[std]{{a*c}x^2+{a*d}x+{b*c}x+{b*d}}\\\\&amp;=&amp;\\simplify[std]{{a*c}x^2+{(a*d+b*c)}x+{b*d}}\\end{eqnarray*}\\]</p>\n  <p>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n      <p>$\\simplify[std]{({a}x+{b})({c}x+{d})}=\\;$[[0]].</p>\n        <p>Your answer should be a quadratic in $x$ and should not include any brackets.</p>\n        <p>You can click on Show steps for more information, but you will lose one mark if you do so.</p>\n      ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Do not include brackets in your answer. Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.</p>", "showStrings": false, "strings": ["("], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "maxlength": {"length": 17, "message": "<p>Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.</p>", "partialCredit": 0}, "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "musthave": {"message": "<p>Input your answer as a quadratic in $x$, in the form $ax^2+bx+c$ for appropriate integers $a,\\;b,\\;c$.</p>", "showStrings": false, "strings": ["x^2"], "partialCredit": 0}, "scripts": {}, "answer": "{a*c}x^2+{b*c+a*d}x+{b*d}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1], "answersimplification": "std"}], "steps": [{"prompt": "\n          <p>There are many ways to expand an expression such as $(ax+b)(cx+d)$.</p>\n          <p>One way:</p>\n          <p>\\[\\begin{eqnarray*} (ax+b)(cx+d)&amp;=&amp;ax(cx+d)+b(cx+d)\\\\&amp;=&amp;acx^2+adx+bcx+bd\\\\&amp;=&amp;acx^2+(ad+bc)x+bd\\end{eqnarray*}\\]</p>\n          ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "<p>Expand the following to give a quadratic in $x$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..9 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(2..9 except [0,c])", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"description": "<p>Expand $(ax+b)(cx+d)$.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Simplify Algebraic Expressions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic manipulation", "expanding brackets", "simplification", "simplifying an expression"], "advice": "\n  <p>Expanding the brackets we have:</p>\n  <p>\\[\\begin{eqnarray*}\\simplify[std]{({a}x+{b})({c}x+{d})-({a}x+{d})({c}x+{b})}&amp;=&amp;(\\simplify[std]{{a*c}x^2+{b*c+a*d}x+{b*d}})-(\\simplify[std]{{a*c}x^2+{b*a+c*d}x+{b*d}})\\\\&amp;=&amp;\\simplify[std]{{b*c+a*d}x-{b*a+c*d}x}\\\\&amp;=&amp;\\var{(a-c)*(d-b)}x\\end{eqnarray*}\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>Simplify:</p>\n      <p>$\\simplify[std]{({a}x+{b})({c}x+{d})-({a}x+{d})({c}x+{b})}=\\;$[[0]]</p>\n      <p>Do not include brackets in your answer.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Do not include brackets in your answer.</p>", "showstrings": false, "strings": ["("], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{(a-c)*(d-b)}*x", "type": "jme", "maxlength": {"length": 6.0, "message": "<p>You can simplify the expression further.</p>", "partialcredit": 0.0}}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Simplify the following expression.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-6..6 except 0)", "name": "a"}, "c": {"definition": "random(-6..6 except [0,a])", "name": "c"}, "b": {"definition": "random(1..9 except a)", "name": "b"}, "d": {"definition": "random(1..9 except c)", "name": "d"}}, "metadata": {"notes": "\n    \t\t<p><strong>18/08/2012:</strong></p>\n    \t\t        <p>Added tags.</p>\n    \t\t        <p>Added description.</p>\n    \t\t", "description": "<p>Simplify $(ax+b)(cx+d)-(ax+d)(cx+b)$. Answer is a multiple of $x$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Simplify Algebraic Expressions: 2 unknowns", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic manipulation", "expanding brackets", "simplification", "simplifying an expression"], "advice": "\n  <p>Expanding the brackets we have:</p>\n    <p>\\[\\begin{eqnarray*}\\simplify[std]{({a}x+{b}y)({c}x+{d}y)-({a}x+{d}y)({c}x+{b}y)}&amp;=&amp;(\\simplify[std]{{a*c}x^2+{b*c+a*d}x*y+{b*d}y^2})-(\\simplify[std]{{a*c}x^2+{b*a+c*d}x*y+{b*d}y^2})\\\\&amp;=&amp;\\simplify[std]{{b*c+a*d}x*y-{b*a+c*d}x*y}\\\\&amp;=&amp;\\var{(a-c)*(d-b)}xy\\end{eqnarray*}\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>Simplify:</p>\n        <p>$\\simplify[std]{({a}x+{b}y)({c}x+{d}y)-({a}x+{d}y)({c}x+{b}y)}=\\;$[[0]]</p>\n        <p>Do not include brackets in your answer.</p>\n        <p>Input $xy$ as $x*y$.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Do not include brackets in your answer.</p>", "showstrings": false, "strings": ["("], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{(a-c)*(d-b)}*x*y", "type": "jme", "maxlength": {"length": 7.0, "message": "<p>You can simplify the expression further.</p>", "partialcredit": 0.0}}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Simplify the following expression.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-6..6 except 0)", "name": "a"}, "c": {"definition": "random(-6..6 except [0,a])", "name": "c"}, "b": {"definition": "random(1..9 except a)", "name": "b"}, "d": {"definition": "random(1..9 except [c,b])", "name": "d"}}, "metadata": {"notes": "\n    \t\t    \t\t<p><strong>18/08/2012:</strong></p>\n    \t\t    \t\t        <p>Added tags.</p>\n    \t\t    \t\t        <p>Added description.</p>\n    \t\t    \n    \t\t", "description": "<p>Simplify $(ax+by)(cx+dy)-(ax+dy)(cx+by)$. Answer is a multiple of $xy$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Expansion of three brackets: (Video)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic manipulation", "expansion of brackets", "expansion of the product of three linear terms", "video"], "advice": "\n  <p>Using the method given by Show steps we first multiply out the first two brackets:</p>\n    <p>\\[\\begin{eqnarray*}\\simplify[std]{ ({a}y+{b})({c}y+{d})}&amp;=&amp;\\simplify[std]{{a}y*({c}y+{d})+{b}({c}y+{d})}\\\\&amp;=&amp;\\simplify[std]{{a*c}y^2+{a*d}y+{b*c}y+{b*d}}\\\\&amp;=&amp;\\simplify[std]{{a*c}y^2+{(a*d+b*c)}y+{b*d}}\\end{eqnarray*}\\]</p>\n    <p>And then multiply this by the third bracket:</p>\n    <p>\\[\\begin{eqnarray*}\\simplify[std]{({a}y+{b})({c}y+{d})({p}y+{q})}&amp;=&amp;\\simplify[std]{({a*c}y^2+{(a*d+b*c)}y+{b*d})({p}y+{q})}\\\\&amp;=&amp;\\simplify[std]{{a*c}y^2({p}y+{q})+{(a*d+b*c)}y*({p}y+{q})+{b*d}({p}y+{q})}\\\\&amp;=&amp;\\simplify[std]{{a*c*p}*y^3 +{a*c*q}*y^2+{p*(a*d+b*c)}y^2+{q*(a*d+b*c)}y+{b*d*p}y+{b*d*q}}\\\\&amp;=&amp;\\simplify[std]{{a*c*p}y^3+{a*c*q+a*d*p+p*b*c}y^2+{a*d*q+b*c*q+b*d*p}y+{b*d*q}}\\end{eqnarray*}\\]</p>\n    <p>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "<p>$\\simplify[std]{({a}y+{b})({c}y+{d})({p}y+{q})}=\\;$[[0]].</p>\n        <p>Your answer should be a cubic in $y$ and should not include any brackets.</p>\n        <p>You can click on Show steps for more information, but you will lose one mark if you do so.</p>\n        <p>There is a video in Show steps which expands three brackets, but uses the variable $x$ rather than $y$.</p>", "gaps": [{"notallowed": {"message": "<p>Do not include brackets in your answer. Input your answer as a cubic in $y$, in the form $ay^3+by^2+cy+d$ for appropriate integers $a,\\;b,\\;c,\\;d$.</p>", "showstrings": false, "strings": ["(", "yy", "y*y"], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{a*c*p}y^3+{a*c*q+a*d*p+p*b*c}y^2+{a*d*q+b*c*q+b*d*p}y+{b*d*q}", "type": "jme"}], "steps": [{"prompt": "<p>There are many ways to expand an expression such as $(ay+b)(cy+d)(py+q)$.</p>\n            <p>One way is to expand the first two brackets, and then multiply the resulting quadratic in $y$ by $py+q$.</p>\n            <p>Hence:</p>\n            <p>\\[\\begin{eqnarray*} (ay+b)(cy+d)&amp;=&amp;ay(cy+d)+b(cy+d)\\\\&amp;=&amp;acy^2+ady+bcy+bd\\\\&amp;=&amp;acy^2+(ad+bc)y+bd\\end{eqnarray*}\\]</p>\n            <p>and then work out $(acy^2+(ad+bc)y+bd)(py+q)$.</p>\n            <p>See this video for more help:</p>\n            <p><iframe style=\"display: block; margin-left: auto; margin-right: auto;\" src=\"https://player.vimeo.com/video/8169375\" frameborder=\"0\" width=\"640\" height=\"480\"></iframe></p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "<p>Expand the following to give a cubic in $y$.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..5)", "name": "a"}, "c": {"definition": "random(2..5)", "name": "c"}, "b": {"definition": "random(-9..9 except [0,a])", "name": "b"}, "d": {"definition": "random(-9..9 except [0,c])", "name": "d"}, "q": {"definition": "random(-3..3 except [0,b,d])", "name": "q"}, "p": {"definition": "random(1..3 except [a,c])", "name": "p"}}, "metadata": {"notes": "\n                            \t\t<p><strong>16/08/2012:</strong></p>\n                            \t\t        <p>Added tags.</p>\n                            \t\t        <p>Added description.</p>\n                            \t\t", "description": "<p>Expand $(ay+b)(cy+d)(py+q)$.</p>\n                        <p>Includes a video expanding three brackets, however uses the variable $x$ rather than $y$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Deirdre Casey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/681/"}], "extensions": [], "custom_part_types": [], "resources": []}