// Numbas version: exam_results_page_options {"name": "Solve equations which include a single even power (e.g. x^even=blah)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solve equations which include a single even power (e.g. x^even=blah)", "tags": [], "metadata": {"description": "

Questions to test if the student knows the inverse of an even power (and how to solve equations that contain a single power that is even). 

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Please complete the following.

", "advice": "

Suppose you wanted to solve $x^2=4$, that is, you wanted the number that squares to give four. The number $2$ comes to mind, however, $-2$ is also a solution since $(-2)^2=(-2)\\times(-2)=4$.  Recall the product of two negatives is a positive, so the product of any even number of negative numbers is positive. This means when we find a positive solution to an equation like $x^4=10\\,000$ there will also be a negative solution, in particular, the solution to $x^4=10\\,000$ would be $x=\\pm\\sqrt[4]{10\\,000}=\\pm10$.

\n

\n

a) Since the power in $x^\\var{intpower}=\\var{intrhs}$ is even we will take the plus or minus $\\var{intpower}$nd rd th  root to get two solutions.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$x^\\var{intpower}$$=$$\\var{intrhs}$ 
 
$\\sqrt[\\var{intpower}]{x^\\var{intpower}}$$=$$\\pm\\sqrt[\\var{intpower}]{\\var{intrhs}}$
 
$x$$=$$\\pm\\var{intsoln}$
\n

That is, $x$ equals $-\\var{intsoln}$ or $\\var{intsoln}$.

\n

\n

b) Given $\\simplify{{bxcoeff}y^{bpower}+{bb}}=\\var{bc}$, we can rearrange the equation to get $y^\\var{bpower}$ by itself and then we can take the plus or minus $\\var{bpower}$nd rd th  root to get $y$ by itself.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{bxcoeff}y^{bpower}+{bb}}$$=$$\\var{bc}$ 
 
$\\simplify{{bxcoeff}y^{bpower}}$$=$$\\simplify[basic]{{bc}-{bb}}$
 
$\\simplify{{bxcoeff}y^{bpower}}$$=$$\\simplify{{bc-bb}}$
$y^\\var{bpower}$$=$$\\simplify[!basic]{{bc-bb}/{bxcoeff}}$
$y^\\var{bpower}$$=$$\\simplify{{bc-bb}/{bxcoeff}}$
$\\sqrt[\\var{bpower}]{y^\\var{bpower}}$$=$$\\pm\\sqrt[\\var{bpower}]{\\var{brhs}}$
$y$$=$$\\pm\\var{bsoln}$
\n

That is, $y$ equals $-\\var{bsoln}$, or $\\var{bsoln}$.

\n

\n

c) Given $\\simplify{{cxcoeff}z^{cpower}+{cb}}=\\var{cc}$, we can rearrange the equation to get $z^\\var{cpower}$ by itself and then we can take the plus or minus $\\var{cpower}$nd rd th  root to get $z$ by itself.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{cxcoeff}z^{cpower}+{cb}}$$=$$\\var{cc}$ 
 
$\\simplify{{cxcoeff}z^{cpower}}$$=$$\\simplify[basic]{{cc}-{cb}}$
 
$\\simplify{{cxcoeff}z^{cpower}}$$=$$\\simplify{{cc-cb}}$
$z^\\var{cpower}$$=$$\\simplify[!basic]{{cc-cb}/{cxcoeff}}$
$z^\\var{cpower}$$=$$\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}$
$\\sqrt[\\var{cpower}]{z^\\var{cpower}}$$=$$\\pm\\sqrt[\\var{cpower}]{\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$
$z$$=$$\\pm\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}^{1/{cpower}}}$
\n

That is, $z$ equals $-\\sqrt[\\var{cpower}]{\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$, or $\\sqrt[\\var{cpower}]{\\simplify[fractionNumbers]{{(cc-cb)/cxcoeff}}}$.

\n

\n

d) Given $\\displaystyle{\\simplify{((z+{db})^{bpower})/({ddenom})}}=\\var{dc}$, we can rearrange the equation to get $\\simplify{(z+{db})^{bpower}}$ by itself, then we can take the plus or minus $\\var{bpower}$nd rd th  root of both sides to get $\\simplify{z+{db}}$ by itself, and then rearrange to get $z$ by itself.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\displaystyle{\\simplify{((z+{db})^{bpower})/({ddenom})}}$$=$$\\var{dc}$ 
 
$\\simplify{(z+{db})^{bpower}}$$=$$\\simplify[basic]{{dc}*{ddenom}}$
 
$\\simplify{(z+{db})^{bpower}}$$=$$\\var{dc*ddenom}$
$\\sqrt[\\var{bpower}]{\\simplify{(z+{db})^{bpower}}}$$=$$\\pm\\sqrt[\\var{bpower}]{\\var{dc*ddenom}}$
$\\simplify{z+{db}}$$=$$\\pm\\sqrt[\\var{bpower}]{\\var{dc*ddenom}}$
$z$$=$$\\pm\\simplify{root({dc*ddenom},{bpower})-{db}}$
$z$$=$$\\pm\\simplify{{dc*ddenom}^(1/{bpower})-{db}}$
\n

That is, $z$ equals $-\\simplify{root({dc*ddenom},{bpower})-{db}}$, or $\\simplify{root({dc*ddenom},{bpower})-{db}}$.

", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"cxcoeff": {"name": "cxcoeff", "group": "c", "definition": "if(cc-cb>0,random(2..12),random(-12..-2))", "description": "", "templateType": "anything", "can_override": false}, "intrhs": {"name": "intrhs", "group": "a", "definition": "intsoln^intpower\n", "description": "", "templateType": "anything", "can_override": false}, "bpower": {"name": "bpower", "group": "b", "definition": "powers[1]", "description": "", "templateType": "anything", "can_override": false}, "db": {"name": "db", "group": "Ungrouped variables", "definition": "random(-100..100 except -1..1)", "description": "", "templateType": "anything", "can_override": false}, "bc": {"name": "bc", "group": "b", "definition": "bsoln^bpower*bxcoeff+bb", "description": "", "templateType": "anything", "can_override": false}, "ddenom": {"name": "ddenom", "group": "Ungrouped variables", "definition": "random(2..15)", "description": "", "templateType": "anything", "can_override": false}, "powers": {"name": "powers", "group": "a", "definition": "shuffle([2,4,6,8])", "description": "", "templateType": "anything", "can_override": false}, "bb": {"name": "bb", "group": "b", "definition": "random(1..100)", "description": "", "templateType": "anything", "can_override": false}, "dc": {"name": "dc", "group": "Ungrouped variables", "definition": "random(2..100)", "description": "", "templateType": "anything", "can_override": false}, "cc": {"name": "cc", "group": "c", "definition": "random(-100..100)", "description": "", "templateType": "anything", "can_override": false}, "brhs": {"name": "brhs", "group": "b", "definition": "(bc-bb)/bxcoeff", "description": "", "templateType": "anything", "can_override": false}, "cb": {"name": "cb", "group": "c", "definition": "random(-100..100 except [0,cc])", "description": "", "templateType": "anything", "can_override": false}, "bxcoeff": {"name": "bxcoeff", "group": "b", "definition": "random(-3..3 except 0..1)", "description": "", "templateType": "anything", "can_override": false}, "intsoln": {"name": "intsoln", "group": "a", "definition": "switch(intpower=2, random(2..12), intpower=4, random(2..5), intpower=6, random(2..3), 2)", "description": "", "templateType": "anything", "can_override": false}, "intpower": {"name": "intpower", "group": "a", "definition": "powers[0]", "description": "", "templateType": "anything", "can_override": false}, "bsoln": {"name": "bsoln", "group": "b", "definition": "switch(bpower=2, random(2..10), bpower=4, random(2..4), bpower=6, random(2..3), 2)", "description": "", "templateType": "anything", "can_override": false}, "cpower": {"name": "cpower", "group": "c", "definition": "powers[2]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dc", "db", "ddenom"], "variable_groups": [{"name": "a", "variables": ["powers", "intpower", "intrhs", "intsoln"]}, {"name": "b", "variables": ["bpower", "bsoln", "bxcoeff", "bb", "bc", "brhs"]}, {"name": "c", "variables": ["cpower", "cc", "cb", "cxcoeff"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "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": "

If  $x^\\var{intpower}=\\var{intrhs}$, then $x=$ [[0]], or [[1]].

\n

(Please enter the smaller value in the first gap and the larger value on the second gap)

", "gaps": [{"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, "answer": "{-intsoln}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"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, "answer": "{intsoln}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "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": "

If  $\\simplify{{bxcoeff}y^{bpower}+{bb}}=\\var{bc}$, then $y=$ [[0]], or [[1]].

\n

(Please enter the smaller value in the first gap and the larger value on the second gap)

", "gaps": [{"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, "answer": "{-bsoln}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"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, "answer": "{bsoln}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "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": "

For this question input the exact value by using a fractional power to indicate a root. For example, if the answer was $\\sqrt[3]{\\frac{35}{11}}$, then enter  (35/11)^(1/3).

\n

\n

If  $\\simplify{{cxcoeff}z^{cpower}+{cb}}=\\var{cc}$, then $z=$ [[0]], or [[1]].

\n

(Please enter the smaller value in the first gap and the larger value on the second gap)

", "gaps": [{"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, "answer": "-({(cc-cb)/cxcoeff})^(1/{cpower})", "answerSimplification": "fractionNumbers", "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": "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, "answer": "({(cc-cb)/cxcoeff})^(1/{cpower})", "answerSimplification": "fractionNumbers", "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": []}], "sortAnswers": false}, {"type": "gapfill", "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": "

For this question input the exact value by using a fractional power to indicate a root. For example, if the answer was $\\sqrt[3]{\\frac{35}{11}}$, then enter  (35/11)^(1/3).

\n

If  $\\displaystyle{\\simplify{((z+{db})^{bpower})/({ddenom})}}=\\var{dc}$, then $z=$ [[0]], or [[1]].

\n

(Please enter the smaller value in the first gap and the larger value on the second gap)

", "gaps": [{"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, "answer": "-({dc*ddenom})^(1/{bpower})-{db}", "answerSimplification": "basic", "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": "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, "answer": "({dc*ddenom})^(1/{bpower})-{db}", "answerSimplification": "basic", "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": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}