{"@context":"https://schema.org/","@type":"Quiz","name":"

$(-1+\\sqrt{-3})^{5}+(-1-\\sqrt{-3})^{5}$

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"NET"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"NET","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/1-sqrt-3-5-1-sqrt-3-5"}],"about":{"@type":"Thing","name":"NET, NUST Entry Test, Mathematics, Quadratic Equations, Quiz"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["NET","NUST Entry Test","Mathematics","Quadratic Equations","Quiz"],"name":"

$(-1+\\sqrt{-3})^{5}+(-1-\\sqrt{-3})^{5}$

","comment":{"@type":"Comment","text":"

$(-1+\\sqrt{-3})^{5}+(-1-\\sqrt{-3})^{5}$

Multiplying and dividing both terms by $2$

$=\\left(2\\cdot\\cfrac{(-1+\\sqrt{-3})}{2}\\right)^{5}+\\left(2\\cdot \\cfrac{(-1-\\sqrt{-3})^{5}}{2}\\right)^{5}$

We know that

$\\omega = \\cfrac{-1+\\sqrt{-3}}{2}, \\quad \\quad\\omega^{2} = \\cfrac{-1+\\sqrt{-3}}{2}$

Hence the above equation becomes:

$=(2\\omega)^{5}+(2\\omega^{2})^{5}$

$=32\\omega^{5}+32\\omega^{10}$

$=32(\\omega^{3}\\cdot\\omega^{2}+\\omega^{9}.\\omega)$

$=32(1\\cdot\\omega^{2}+(\\omega^{3})^{3}.\\omega) \\quad\\quad\\quad\\quad\\quad \\because \\omega^{3}=1$

$=32(\\omega^{2}+(1)^{3}.\\omega)$

$=32(\\omega^{2}+\\omega)$

$=32(-1) \\quad\\quad\\quad\\quad\\quad \\because \\omega +\\omega^{2} =-1$

$=-32$

"},"encodingFormat":"text/markdown","text":"

$(-1+\\sqrt{-3})^{5}+(-1-\\sqrt{-3})^{5}$

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$40$

"},{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$38$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$30$

"}],"acceptedAnswer":{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$-32$

","comment":{"@type":"Comment","text":"

$(-1+\\sqrt{-3})^{5}+(-1-\\sqrt{-3})^{5}$

Multiplying and dividing both terms by $2$

$=\\left(2\\cdot\\cfrac{(-1+\\sqrt{-3})}{2}\\right)^{5}+\\left(2\\cdot \\cfrac{(-1-\\sqrt{-3})^{5}}{2}\\right)^{5}$

We know that

$\\omega = \\cfrac{-1+\\sqrt{-3}}{2}, \\quad \\quad\\omega^{2} = \\cfrac{-1+\\sqrt{-3}}{2}$

Hence the above equation becomes:

$=(2\\omega)^{5}+(2\\omega^{2})^{5}$

$=32\\omega^{5}+32\\omega^{10}$

$=32(\\omega^{3}\\cdot\\omega^{2}+\\omega^{9}.\\omega)$

$=32(1\\cdot\\omega^{2}+(\\omega^{3})^{3}.\\omega) \\quad\\quad\\quad\\quad\\quad \\because \\omega^{3}=1$

$=32(\\omega^{2}+(1)^{3}.\\omega)$

$=32(\\omega^{2}+\\omega)$

$=32(-1) \\quad\\quad\\quad\\quad\\quad \\because \\omega +\\omega^{2} =-1$

$=-32$

"},"answerExplanation":{"@type":"Comment","text":"

$(-1+\\sqrt{-3})^{5}+(-1-\\sqrt{-3})^{5}$

Multiplying and dividing both terms by $2$

$=\\left(2\\cdot\\cfrac{(-1+\\sqrt{-3})}{2}\\right)^{5}+\\left(2\\cdot \\cfrac{(-1-\\sqrt{-3})^{5}}{2}\\right)^{5}$

We know that

$\\omega = \\cfrac{-1+\\sqrt{-3}}{2}, \\quad \\quad\\omega^{2} = \\cfrac{-1+\\sqrt{-3}}{2}$

Hence the above equation becomes:

$=(2\\omega)^{5}+(2\\omega^{2})^{5}$

$=32\\omega^{5}+32\\omega^{10}$

$=32(\\omega^{3}\\cdot\\omega^{2}+\\omega^{9}.\\omega)$

$=32(1\\cdot\\omega^{2}+(\\omega^{3})^{3}.\\omega) \\quad\\quad\\quad\\quad\\quad \\because \\omega^{3}=1$

$=32(\\omega^{2}+(1)^{3}.\\omega)$

$=32(\\omega^{2}+\\omega)$

$=32(-1) \\quad\\quad\\quad\\quad\\quad \\because \\omega +\\omega^{2} =-1$

$=-32$

"}}},"url":"https://classnotes.xyz/mcqs/1-sqrt-3-5-1-sqrt-3-5"}

$(-1+\sqrt{-3})^{5}+(-1-\sqrt{-3})^{5}$