find all the fourth roots of -8 in Cartesian form.
simplify as much as possilbe
z4=−8z1, z2, z3, z4= ?z4=(−2)⋅4|±√z2=±2⋅√−2z2=±2⋅√(−1)⋅2z2=±2⋅√2⋅√−1|√−1=iz2=±2⋅√2⋅i
Solutions z1, z2, z3, z4:
(1)z21=+2⋅√2⋅i(2)z22=(−z1)2=+2⋅√2⋅iorz2=−z1(3)z23=−2⋅√2⋅i(4)z24=(−z3)2=−2⋅√2⋅iorz4=−z3
z1(1)z21=+2⋅√2⋅i|z1=a+b⋅i(a+b⋅i)2=+2⋅√2⋅ia2+2abi+b2i2=+2⋅√2⋅i|i2=−1a2+2abi−b2=+2⋅√2⋅ia2−b2⏟Re=0+2abi⏟Im=+2⋅√2⋅i=+2⋅√2⋅ia2−b2=0a2=b22abi=+2√2⋅iab=√2b=√2aa2=b2a2=(√2a)2a2=2a2a4=2a=4√2b=√2ab=√24√2b=212214b=212−14b=214b=4√2z1=a+biz1=4√2+4√2⋅iz1=4√2⋅(1+i)
z2(2)z2=−z1z2=−4√2⋅(1+i)
z3(3)z23=−2⋅√2⋅i|z3=a+b⋅i(a+b⋅i)2=−2⋅√2⋅ia2+2abi+b2i2=−2⋅√2⋅i|i2=−1a2+2abi−b2=−2⋅√2⋅ia2−b2⏟Re=0+2abi⏟Im=−2⋅√2⋅i=−2⋅√2⋅ia2−b2=0a2=b22abi=−2√2⋅iab=−√2b=−√2aa2=b2a2=(−√2a)2a2=2a2a4=2a=4√2b=−√2ab=−√24√2b=−212214b=−212−14b=−214b=−4√2z3=a+biz3=4√2−4√2⋅iz3=4√2⋅(1−i)
z4(4)z4=−z3z4=−4√2⋅(1−i)
z1=4√2⋅(1+i)z2=−4√2⋅(1+i)z3=4√2⋅(1−i)z4=−4√2⋅(1−i)
