Y by 62861, jhu08, Adventure10, Mango247
Find ![$ \sqrt [3] {cos \frac{2\pi}{7}}+\sqrt [3] {cos \frac{4\pi}{7}}+\sqrt [3] {cos \frac{8\pi}{7}}$](//latex.artofproblemsolving.com/d/e/9/de9d8db03700f9b9ca729b6c779e94d733559c15.png)
![$ \sqrt [3] {cos \frac{2\pi}{7}}+\sqrt [3] {cos \frac{4\pi}{7}}+\sqrt [3] {cos \frac{8\pi}{7}}$](http://latex.artofproblemsolving.com/d/e/9/de9d8db03700f9b9ca729b6c779e94d733559c15.png)
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expand(subst ([x=(135*u^3-5-sqrt(5*(729*u^6+270*u^3+192*u+5)))/(90*u)], 81*x^5+135*u*x^3+15*x^2+45*u^2*x+5*u+19));
ratsimp(%);
allroots(531441*u^12+531441*u^11+354294*u^10+688905*u^9+885735*u^8+509571*u^7+
94041*u^6+96714*u^5+106515*u^4+36180*u^3+10044*u^2+1029*u+71);
[u=0.08873583372698096*%i-0.06285164678937094,u=-0.08873583372698096*%i-0.06285164678937094,
u=0.249818724727736*%i-0.1038150198772957,
u=-0.249818724727736*%i-.1038150198772957,u=0.4141258170774749*%i+0.40187021328863,
u=0.40187021328863-0.4141258170774749*%i,u=0.04493113182149037*%i-0.7343941915363165,
u=-0.04493113182149037*%i-0.7343941915363165,u=0.6072801786518509*%i-0.5685368799552993,
u=-0.6072801786518509*%i-.5685368799552993,u=0.9566536050241474*%i+0.5677275248696524,
u=0.5677275248696524-0.9566536050241474*%i]
allroots(531441*u^12-354294*u^11+59049*u^10+492075*u^9-623295*u^8+236196*u^7+305451*u^6-360126*u^5+
336150*u^4-178335*u^3+109719*u^2-16851*u+19531);
[u = 0.4301615523642846 %i - 0.1842082400516285,
u = (- 0.4301615523642846 %i) - 0.1842082400516285,
u = 0.5973695492100051 %i - 0.001309564037543536,
u = (- 0.5973695492100051 %i) - 0.001309564037543536,
u = 0.5364026821017057 %i + 0.5180417823368586,
u = 0.5180417823368586 - 0.5364026821017057 %i,
u = 0.3315150891954592 %i + 0.81690633083384,
u = 0.81690633083384 - 0.3315150891954592 %i,
u = 0.9665642344659907 %i + 0.2055111206314012,
u = 0.2055111206314012 - 0.9665642344659907 %i,
u = 0.2658544600145461 %i - 1.021608096379594,
u = (- 0.2658544600145461 %i) - 1.021608096379594]
ratsimp(ratsimp(expand(subst ([x=(135*u^3-5+sqrt(5*(729*u^6+270*u^3+192*u+5)))/(90*u)], 81*x^5+135*u*x^3+15*x^2+45*u^2*x+5*u+19))));
solve(-24912826983452529*u^30-25164471700457100*u^27-10438595668337760*u^25-2541865828329000*u^24-1656919947355200*u^22+
203976887458500*u^21-187449529397760*u^20-9298091736000*u^19+35836395232500*u^18+255981631098600*u^17+5337793404000*u^16+
97391914860870*u^15+37751974317000*u^14-1211685480000*u^13+26705675099887200*u^12-671209983000*u^11+10171979570973780*u^10+
3529883016229500*u^9-75582720000*u^8+1506164582102400*u^7+78438320892000*u^6+720019630933128*u^5-8264970432000*u^2-860934420000*
u+25736373575697375,u);
[[0=102521921742603*u^30+103557496709700*u^27+42957183820320*u^25+10460353203000*u^24+6818600606400*u^22-839411059500*
u^21+771397240320*u^20+38263752000*u^19-147474877500*u^18-1053422350200*u^17-21966228000*u^16-400789773090*u^15-155357919000*u^14+
4986360000*u^13-109899897530400*u^12+2762181000*u^11-41859998234460*u^10-14526267556500*u^9+311040000*u^8-6198208156800*u^7-
322791444000*u^6-2963043748696*u^5+34012224000*u^2+3542940000*u-105911002369125]=0]
allroots(102521921742603*u^30+103557496709700*u^27+42957183820320*u^25+10460353203000*u^24+6818600606400*u^22-839411059500*
u^21+771397240320*u^20+38263752000*u^19-147474877500*u^18-1053422350200*u^17-21966228000*u^16-400789773090*u^15-155357919000*u^14+
4986360000*u^13-109899897530400*u^12+2762181000*u^11-41859998234460*u^10-14526267556500*u^9+311040000*u^8-6198208156800*u^7-
322791444000*u^6-2963043748696*u^5+34012224000*u^2+3542940000*u-105911002369125,u);
u = 0.6701030312889549 - 0.7418603072941804 %i,
u = 0.6936763996501353 %i - 0.6394643083013833,
u = (- 0.6936763996501353 %i) - 0.6394643083013833,
u = 0.8693833939058644 %i - 0.5003277841706911,
u = (- 0.8693833939058644 %i) - 0.5003277841706911, u = 1.003024944024268,
u = 0.9515836439685664 %i + 0.308918723378388,
u = 0.308918723378388 - 0.9515836439685664 %i,
u = 0.5896639595395451 %i - 0.8091698531500543,
u = (- 0.5896639595395451 %i) - 0.8091698531500543,
u = 0.9340675629015904 %i - 0.2681691050314659,
u = (- 0.9340675629015904 %i) - 0.2681691050314659,
u = 0.9930125376811316 %i + 0.1466433765516465,
u = 0.1466433765516465 - 0.9930125376811316 %i,
u = 0.9950264234389671 %i - 0.1076485022948794,
u = (- 0.9950264234389671 %i) - 0.1076485022948794,
u = 0.3220753007654913 %i - 0.9341384895486041,
u = (- 0.3220753007654913 %i) - 0.9341384895486041,
u = 0.2386333647302399 %i + 0.9080121402521805,
u = 0.9080121402521805 - 0.2386333647302399 %i, u = - 1.132184434639223,
u = 0.9072105915610396 %i + 0.5644742490679983,
u = 0.5644742490679983 - 0.9072105915610396 %i,
u = 0.6763316638574209 %i + 0.7886132313994743,
u = 0.7886132313994743 - 0.6763316638574209 %i,
u = 0.2073934966713666 %i - 0.9781321310923142,
u = (- 0.2073934966713666 %i) - 0.9781321310923142,
u = 0.406721388877138 %i + 0.9148651669582274,
u = 0.9148651669582274 - 0.406721388877138 %i]
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