Question

Find all solutions to the equation x4 - 31 = 0. Leave your answers in terms of pi.

Answer 1

According a 2 31. 3 ---(1) Solution: x - 320 x = 3:0 x = (3:16 to de Moivere's formula, complex any number, 2-arib = 8 (caso ti säng) where, &- $2²+62 and do tant be 3-3 (COBA + i sin a here, so 3, and on In General, we con elewsite 3; as cas (2 ka + 7) + i sin (aka a where kis any integer, Now, Again from de Moi vere's formula If any complex sumber bzyk - pleos 0 + isingen (2) (8gth [cas of In. General case: stows, using equeation izja = levytan (cos (2.*** 0)+i4nente) where, k=0,1,2,....na Now, from equations 0 ① and W, we get cos (2F t i sin. reation and o OY

for k=0, 12, 4) 5 kr 0,1,2,3 ) ( + + 2 k2011, 23 (condition from equation (ing] * 200k ( ces (24:13). I min (*** 9) + i sin (KA . All solution to the given equation is for ko, %. = 1856 ( cas of + i sin ] fon kal, x, = leyla ( cos(4-1) + 4 (-1)] for k=2, x₃ = (3 ja [cas (7 + 7) + i sin ( + 7 for k=3, %= (33" (cos (3*+ 5) dinin (24+)] +

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