x^6 = 64, taking z = 2e^it gives 64e^i(6t) = 64 e^i(6t) = 1 Then the argument 6t has to be some 2kpi, t=0,2pi/6,4pi/6,6pi/6,8pi/6,10pi/6 gives us our 6 solutions in exponential form. t=0 gives z=2 t=pi/3 gives z=sqrt(3) + i t=2pi/3 gives z=-sqrt(3) + i t=pi gives z=-2 t=4pi/3 gives z=-sqrt(3) - i t=5pi/3 gives z=sqrt(3) - i
2
I spent too much time programming with assembler to be scared of 2s powers
Just do
³√x^6 = ³√64
x² = 4
√x² = √4
x = 2
x^6 = 64, taking z = 2e^it gives
64e^i(6t) = 64
e^i(6t) = 1
Then the argument 6t has to be some 2kpi,
t=0,2pi/6,4pi/6,6pi/6,8pi/6,10pi/6
gives us our 6 solutions in exponential form.
t=0 gives z=2
t=pi/3 gives z=sqrt(3) + i
t=2pi/3 gives z=-sqrt(3) + i
t=pi gives z=-2
t=4pi/3 gives z=-sqrt(3) - i
t=5pi/3 gives z=sqrt(3) - i
64 is 8^2
8 is 2^3
(2^3)^2 is 2^6
x^6 = 2^6
x=2