find all the specified root of certain real or complex numbe

find all the specified root of certain real or complex numbers.

-Find all 8th roots of unity.

-Find all quartic roots 256.

-Find all square roots 4i.

3-find all (real and complex) solutions of the given equation.

x^6 +7x^3 8=0.

Solution

a) Use De Moivre\'s Theorem to find roots of units:

1 = cos 0 + i sin 0
= cos(2 pi k) + i sin(2 pi k), with k being any integer.

So, by De Moivre\'s Theorem, the eight roots of 1 are : z= 1^1/8
cos(2 pi k/8) + i sin(2 pi k/8), with k = 0,1,2,...,8

More specifically, the eight 8th roots of unity are
k = 0: cos 0 + i sin 0 = 1
k = 1: cos(pi/4) + i sin(pi/4) = sqrt(2)/2 + i sqrt(2)/2
k = 2: cos(pi/2) + i sin(pi/2) = i
k = 3: cos(3pi/4) + i sin(3pi/4) = -sqrt(2)/2 + i sqrt(2)/2
k = 4: cos(pi) + i sin(pi) = -1
k = 5: cos(5pi/4) + i sin(5pi/4) = sqrt(2)/2 - i sqrt(2)/2
k = 6: cos(3pi/2) + i sin(3pi/2) = -i
k = 7: cos(7pi/4) + i sin(7pi/4) = -sqrt(2)/2 - i sqrt(2)/2

b) quartic roots 256

z= (-256)^1/4

In polar form : r= 256[-cospi + i*sinpi]

z = {256[cos(pi +2kpi) +sin(pi +2kpi)]}^1/4

= 4[cos(pi+2kpi)/4 +isin(pi+2kpi)/4]

k =0 ;z1 = 4(cospi/4 +isinpi/4) = 2.82 +i*2.82

k=1 z2 = 4(cos135 +isin135) =4( -0.707 +i0.707) = -2.82 +i*2.82

k=2 z3 = 4(cos225 +isin225) = -2.82 -2.82*i

k=3 z4 = 4(cos315 +isin315) = 2.82 -i*2.82

z = (-4i)^1/2

Polar form : z = 4e^i(270)

z= { 4[cos3pi/2 +isin3pi/2]}^1/2

= {2(cos(3pi/2+2kpi)/2

Demovries theorem z= [2(cos(3pi/2+2kpi)/2 +isin(3pi/2+2kpi)/2)

k=0 z1 = {2[cos3pi/4 +isin3pi/4] } = -1.41 + i*1.41

k=1 z2 = 2[cos(3pi/2+2pi)/2 +isin(3pi/2+2pi)/2] 3/4 +1

= z2 = 2[cos7pi/4 +isin7pi/4] = 1.414 -i*1.414