// Numbas version: exam_results_page_options
{"name": "1.2.5. Fractional Indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1.2.5. Fractional Indices", "tags": [], "metadata": {"description": "<p>Part of HELM Book 1.2</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<h3>Fractional Indices</h3>\n<p>So far we have used indices that are whole numbers. We now consider fractional powers. Consider the expression $\\displaystyle{\\left(16^{\\frac{1}{2}}\\right)^2}$. Using the third law of indices, $(a^m)^n=a^{mn}$, we can write</p>\n<p style=\"text-align: center;\">\\( \\left( 16^{\\frac{1}{2}} \\right)^2 = 16^{ \\frac{1}{2} \\times 2 } = 16^1 = 16. \\)</p>\n<p>So $\\displaystyle{16^{\\frac{1}{2}}}$ is a number which when squared equals $16$, that is $4$ or $&minus;4$. In other words $\\displaystyle{16^{\\frac{1}{2}}}$ is a square root of $16$. There are always two square roots of a non-zero positive number, and we write $\\displaystyle{16^{\\frac{1}{2}}=\\pm 4}$</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h4 style=\"text-align: center;\">Key Point 10</h4>\n<p style=\"text-align: center;\">In general $\\displaystyle{a^{\\frac{1}{2}}}$ is a square root of $a$, $a\\geq 0$</p>\n</div>\n<p></p>\n<p>Similarly</p>\n<p style=\"text-align: center;\">\\( \\left(8^{\\frac{1}{3}}\\right)^3=8^{\\frac{1}{3}\\times 3}=8^1=8 \\)</p>\n<p>so that $\\displaystyle{8^{\\frac{1}{3}}}$ is a number which when cubed equals $8$. Thus $\\displaystyle{8^{\\frac{1}{3}}}$ is the cube root of $8$, that is $\\sqrt[3]{8}$, namely $2$. Each number has only one cube root, and so \\( 8^{\\frac{1}{3}} = 2 \\)</p>\n<p>In general</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h4 style=\"text-align: center;\">Key Point 11</h4>\n<p style=\"text-align: center;\">In general $\\displaystyle{a^{\\frac{1}{3}}}$ is the cube root of $a$</p>\n</div>\n<p></p>\n<p>More generally we have</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h4 style=\"text-align: center;\">Key Point 12</h4>\n<p style=\"text-align: center;\">The $n$th root of $a$ is denoted by $\\displaystyle{a^{\\frac{1}{n}}}$</p>\n<p style=\"text-align: center;\">When $a&lt;0$ the $n$th root only exists if $n$ is odd.</p>\n<p style=\"text-align: center;\">When $a&gt;0$ the positive $n$th root is denoted by $\\sqrt[n]{a}$.</p>\n<p style=\"text-align: center;\">If $a&lt;0$ the negative $n$th root is $-\\sqrt[n]{|a|}$.</p>\n</div>\n<p></p>\n<p>Your calculator will be able to evaluate fractional powers, and roots of numbers. Check that you can obtain the results of the following Examples on your calculator, but be aware that calculators normally give only one root when there may be others.</p>\n<p><strong>Example 22</strong></p>\n<p style=\"padding-left: 40px;\">Evaluate (a) $\\displaystyle{144^{\\frac{1}{2}}}$,$\\quad$(b)$\\displaystyle{125^{\\frac{1}{3}}}$</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">(a) $\\displaystyle{144^{\\frac{1}{2}}}$ is a square root of $144$, that is $\\pm 12$.</p>\n<p style=\"padding-left: 40px;\">(b) Noting that $5^3=125$, we see that $\\displaystyle{125^{\\frac{1}{3}}}=\\sqrt[3]{125}=5$</p>\n<p><strong>Example 23</strong></p>\n<p style=\"padding-left: 40px;\">Evaluate (a) $\\displaystyle{32^{\\frac{1}{5}}}$,$\\quad$ (b) $\\displaystyle{32^{\\frac{2}{5}}}$,$\\quad$ (c)$\\displaystyle{8^{\\frac{2}{3}}}$</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">(a) $\\displaystyle{32^{\\frac{1}{5}}}$ is the $5$th root of $32$, that is $\\sqrt[5]{32}$. Now $2^5=32$ and so $\\sqrt[5]{32}=2$.</p>\n<p style=\"padding-left: 40px;\">(b) Using the third law of indices we can write $\\displaystyle{32^{\\frac{2}{5}} = 32^{\\left(2\\times\\frac{1}{5}\\right)}=(32^{\\frac{1}{5}})^2}$.Thus</p>\n<p style=\"padding-left: 40px;\">\\( 32^{\\frac{2}{5}} = ((32)^{\\frac{1}{5}})^2=2^2=4 \\)</p>\n<p style=\"padding-left: 40px;\">(c) Note that $\\displaystyle{8^{\\frac{1}{3}}=2}$. Then&nbsp;</p>\n<p style=\"padding-left: 40px;\">\\( 8^{\\frac{2}{3}} = 8^{\\left( 2\\times\\frac{1}{3}\\right)}= (8^{\\frac{1}{3}})^2 = 2^2=4 \\)</p>\n<p style=\"padding-left: 40px;\">Note the following alternatives:</p>\n<p style=\"padding-left: 40px;\">\\( 8^{\\frac{2}{3}} = (8^{\\frac{1}{3}})^2 = (8^2)^{\\frac{1}{3}} \\)</p>\n<p><strong>Example 24</strong></p>\n<p style=\"padding-left: 40px;\">Write the following as a simple power with a single index:</p>\n<p style=\"padding-left: 40px;\">(a) $\\displaystyle{\\sqrt{x^5}}$,$\\quad$ (b) $\\displaystyle{\\sqrt[4]{x^3}}$</p>\n<p><strong>Solution</strong><strong></strong></p>\n<p style=\"padding-left: 40px;\">(a) $\\displaystyle{\\sqrt{x^5}=(x^5)^{\\frac{1}{2}}}$. Then using the third law of indices we can write this as $\\displaystyle{x^{\\left(5\\times\\frac{1}{2}\\right)}=x^{\\frac{5}{2}}}$.</p>\n<p style=\"padding-left: 40px;\">(b) $\\displaystyle{\\sqrt[4]{x^3} = (x^3)^{\\frac{1}{4}}}$. Using the third law we can write this as $\\displaystyle{ x^{\\left( 3\\times\\frac{1}{4} \\right)} = x^{ \\frac{3}{4} } }$.</p>\n<p><strong>Example 25</strong></p>\n<p style=\"padding-left: 40px;\">Show that $\\displaystyle{ z^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{z}} }$.</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">\\( z^{-\\frac{1}{2}} = \\frac{1}{z^\\frac{1}{2} } = \\frac{1}{\\sqrt{z}} \\)</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q1_expr": {"name": "q1_expr", "group": "question 1", "definition": "latex(\"\\\\frac\\{\" + q1_num+\"\\}\\{\"+q1_den + \"\\}\")", "description": "", "templateType": "anything", "can_override": false}, "q1_num": {"name": "q1_num", "group": "question 1", "definition": "'\\\\sqrt' + q1r1 + '\\{'+q1_letter+'\\}'", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "q1_letter": {"name": "q1_letter", "group": "question 1", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "q1v1": {"name": "q1v1", "group": "question 1", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "q1r1": {"name": "q1r1", "group": "question 1", "definition": "if(q1v1=2,'',\n  if(q1v1=3,'[3]','[4]')\n)", "description": "", "templateType": "anything", "can_override": false}, "q1v2": {"name": "q1v2", "group": "question 1", "definition": "rational(random(1,-1)*random(5,4,3,2,1,1/4,1/3,1/2))", "description": "", "templateType": "anything", "can_override": false}, "q1v3": {"name": "q1v3", "group": "question 1", "definition": "rational(random(1,-1)*random([5,4,3,2,1,1/4,1/3,1/2] except q1v2))", "description": "", "templateType": "anything", "can_override": false}, "q1_den": {"name": "q1_den", "group": "question 1", "definition": "\"(\"+q1_letter + \"^\\{\" + q1v2 + \"\\}\" + q1_letter + \"^\\{\" + q1v3 + \"\\})\"", "description": "", "templateType": "anything", "can_override": false}, "q1_ans": {"name": "q1_ans", "group": "question 1", "definition": "expression(q1_letter+\"^\"+ \"(\" + (1/q1v1 - q1v2 - q1v3) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q2_expr": {"name": "q2_expr", "group": "question 2", "definition": "expression(q2_base+\"^(\"+q2_idx+\")\")", "description": "", "templateType": "anything", "can_override": false}, "q2_base": {"name": "q2_base", "group": "question 2", "definition": "random(2..99)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx": {"name": "q2_idx", "group": "question 2", "definition": "random(-1,1)*(q2_idxnum/q2_idxden)", "description": "", "templateType": "anything", "can_override": false}, "q2_idxden": {"name": "q2_idxden", "group": "question 2", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "q2_idxnum": {"name": "q2_idxnum", "group": "question 2", "definition": "random(1..q2_idxden-1)", "description": "", "templateType": "anything", "can_override": false}, "q3Letter": {"name": "q3Letter", "group": "question 3", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "q3T1idx": {"name": "q3T1idx", "group": "question 3", "definition": "weighted_random([[0,0.5],[random(q3IdxSurds),0.25],[random(q3IdxFracs),0.25]])", "description": "", "templateType": "anything", "can_override": false}, "q3idxSurds": {"name": "q3idxSurds", "group": "question 3", "definition": "[1/2,1/3,1/4]", "description": "", "templateType": "anything", "can_override": false}, "q3idxFracs": {"name": "q3idxFracs", "group": "question 3", "definition": "eval(expression(random(-1,1)+\"*\"+(random(1..5)/random(1..9))))", "description": "", "templateType": "anything", "can_override": false}, "q3T2idx": {"name": "q3T2idx", "group": "question 3", "definition": "random(random(q3IdxSurds),random(q3IdxFracs))", "description": "", "templateType": "anything", "can_override": false}, "q3T3idx": {"name": "q3T3idx", "group": "question 3", "definition": "weighted_random([[0,0.5],[random(q3idxSurds),0.25],[random(q3idxFracs),0.25]])", "description": "", "templateType": "anything", "can_override": false}, "q3T4Idx": {"name": "q3T4Idx", "group": "question 3", "definition": "random(random(q3idxSurds),random(q3idxFracs))", "description": "", "templateType": "anything", "can_override": false}, "q3T1Latex": {"name": "q3T1Latex", "group": "question 3", "definition": "switch(q3T1idx=0,\"\",\n  q3T1idx=1,q3letter,\n  q3T1idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n  q3T1idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n  q3T1idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n  q3letter + \"^\\{\" + q3T1idxStr + \"\\}\"\n  )", "description": "", "templateType": "anything", "can_override": false}, "q3T1idxStr": {"name": "q3T1idxStr", "group": "question 3", "definition": "string(simplify(expression(string(q3T1idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3T2idxStr": {"name": "q3T2idxStr", "group": "question 3", "definition": "string(simplify(expression(string(q3T2idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3T3idxStr": {"name": "q3T3idxStr", "group": "question 3", "definition": "string(simplify(expression(string(q3T3idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3T4idxStr": {"name": "q3T4idxStr", "group": "question 3", "definition": "string(simplify(expression(string(q3T4idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3T2Latex": {"name": "q3T2Latex", "group": "question 3", "definition": "switch(q3T2idx=0,\"\",\n  q3T2idx=1,q3letter,\n  q3T2idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n  q3T2idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n  q3T2idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n  q3letter + \"^\\{\" + q3T2idxStr + \"\\}\"\n  )", "description": "", "templateType": "anything", "can_override": false}, "q3T3Latex": {"name": "q3T3Latex", "group": "question 3", "definition": "switch(q3T3idx=0,\"\",\n  q3T3idx=1,q3letter,\n  q3T3idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n  q3T3idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n  q3T3idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n  q3letter + \"^\\{\" + q3T3idxStr + \"\\}\"\n  )", "description": "", "templateType": "anything", "can_override": false}, "q3T4Latex": {"name": "q3T4Latex", "group": "question 3", "definition": "switch(q3T4idx=0,\"\",\n  q3T4idx=1,q3letter,\n  q3T4idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n  q3T4idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n  q3T4idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n  q3letter + \"^\\{\" + q3T4idxStr + \"\\}\"\n  )", "description": "", "templateType": "anything", "can_override": false}, "q3_num": {"name": "q3_num", "group": "question 3", "definition": "if(q3T1Latex=\"\",q3T2Latex, q3T1Latex + \"\\\\,\" + q3T2Latex)", "description": "", "templateType": "anything", "can_override": false}, "q3_den": {"name": "q3_den", "group": "question 3", "definition": "if(q3T3Latex=\"\",q3T4Latex, q3T3Latex + \"\\\\,\"+ q3T4Latex)", "description": "", "templateType": "anything", "can_override": false}, "q3_expr": {"name": "q3_expr", "group": "question 3", "definition": "latex(\"\\\\frac\\{\"+q3_num+\"\\}\\{\"+q3_den+\"\\}\")", "description": "", "templateType": "anything", "can_override": false}, "q3_ans": {"name": "q3_ans", "group": "question 3", "definition": "expression(q3letter+\"^(\"+ (q3T1idx+q3t2idx-q3t3idx-q3t4idx) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4a_expr": {"name": "q4a_expr", "group": "question 4", "definition": "expression(\"(\" + q4_letter + \"^(\" + q4indices[0] + \"))^(\" + q4indices[1]+\")\")", "description": "", "templateType": "anything", "can_override": false}, "q4b_expr": {"name": "q4b_expr", "group": "question 4", "definition": "expression(q4_letter+\"^(\" + q4indices[2] + \")*\" + q4_letter + \"^(\" + q4indices[3] + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4c_expr": {"name": "q4c_expr", "group": "question 4", "definition": "expression( q4_letter + \"^(\" + q4indices[4] + \")/\" + q4_letter + \"^(\" + q4indices[5] +\")\")", "description": "", "templateType": "anything", "can_override": false}, "q4a_ans": {"name": "q4a_ans", "group": "question 4", "definition": "expression(q4_letter+\"^(\" + (q4indices[0]*q4indices[1]) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4b_ans": {"name": "q4b_ans", "group": "question 4", "definition": "expression(q4_letter+\"^(\" + (q4indices[2]+q4indices[3]) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4c_ans": {"name": "q4c_ans", "group": "question 4", "definition": "expression(q4_letter+\"^(\" + (q4indices[4]-q4indices[5]) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4_letter": {"name": "q4_letter", "group": "question 4", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "q4i_arr": {"name": "q4i_arr", "group": "question 4", "definition": "[1/5,1/4,1/3,1/2,2,3,4,5]", "description": "", "templateType": "anything", "can_override": false}, "q4indices": {"name": "q4indices", "group": "question 4", "definition": "[random(-1,1)*random(q4i_arr),\n  random(-1,1)*random(q4i_arr),\n  random(-1,1)*random(q4i_arr),\n  random(-1,1)*random(q4i_arr),\n  random(-1,1)*random(q4i_arr),\n  random(-1,1)*random(q4i_arr)\n  ]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet"], "variable_groups": [{"name": "question 1", "variables": ["q1_expr", "q1_letter", "q1v1", "q1r1", "q1_num", "q1v2", "q1v3", "q1_den", "q1_ans"]}, {"name": "question 2", "variables": ["q2_expr", "q2_base", "q2_idx", "q2_idxden", "q2_idxnum"]}, {"name": "question 3", "variables": ["q3Letter", "q3idxSurds", "q3idxFracs", "q3T1idx", "q3T1idxStr", "q3T1Latex", "q3T2idx", "q3T2idxStr", "q3T2Latex", "q3T3idx", "q3T3idxStr", "q3T3Latex", "q3T4Idx", "q3T4idxStr", "q3T4Latex", "q3_num", "q3_den", "q3_expr", "q3_ans"]}, {"name": "question 4", "variables": ["q4_letter", "q4i_arr", "q4indices", "q4a_expr", "q4b_expr", "q4c_expr", "q4a_ans", "q4b_ans", "q4c_ans"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": true, "customName": "Task 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Simplify $\\displaystyle{\\var{q1_expr}}$&nbsp;</p>", "stepsPenalty": 0, "steps": [{"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, "prompt": "<p>First, rewrite $\\var{latex(q1_num)}$ using an index and simplify the denominator using the first law of indices.</p>", "answer": "{expression(q1_letter+\"^(1/\"+q1v1+\")/\"+q1_letter+\"^(\"+(q1v2+q1v3)+\")\")}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"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>Finally, use the second law to simplify the result.</p>"}], "answer": "{q1_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "?^? `| ?/?^?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}, {"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><strong>Example 26</strong></p>\n<p>The generalisation of the third law of indices states that $(a^mb^n)^k = a^{mk}b^{nk}$. By<br/>taking $m = 1$, $n = 1$ and $k = \\frac{1}{2}$ show that $\\sqrt{ab}=\\sqrt{a}\\sqrt{b}$.</p>\n<p><br/><b>Solution</b></p>\n<p>Taking $m = 1$, $n = 1$ and $k = \\frac{1}{2}$ gives $(ab)^{\\frac{1}{2}} = a^{\\frac{1}{2}}b^{\\frac{1}{2}}$.</p>\n<p>Taking the case when all these roots are positive we have $\\sqrt{ab}=\\sqrt{a}\\sqrt{b}$.</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h4 style=\"text-align: center;\">Key Point 13</h4>\n<p style=\"text-align: center;\">\\( \\sqrt{ab}=\\sqrt{a}\\sqrt{b},\\qquad a\\geq0,\\; b\\geq0 \\)</p>\n</div>\n<p></p>\n<p>This result often allows answers to be written in alternative forms. For example, we may write $\\sqrt{48}$ as $\\sqrt{3\\times 16} = \\sqrt{3}\\sqrt{16}=4\\sqrt{3}$</p>\n<p>Although this rule works for multiplication we should be aware that it does <strong>not</strong> work for addition or subtraction so that</p>\n<p style=\"text-align: center;\">\\( \\sqrt{a\\pm b} \\neq \\sqrt{a}\\pm \\sqrt{b} \\)</p>"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}]}], "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}