Question #346223

a solution of 2z3+16i=0


1
Expert's answer
2022-05-31T16:23:33-0400
2z3+16i=02z^3+16i=0

z3=8iz^3=-8i

z=2i3z=2\sqrt[3]{-i}

i=cos(π2)+isin(π2)-i=\cos(-\dfrac{\pi}{2})+i\sin(-\dfrac{\pi}{2})

k=0:k=0:


13(cos(π/2+2π(0)3)+isin(π/2+2π(0)3))\sqrt[3]{1}(\cos(\dfrac{-\pi/2+2\pi(0)}{3})+i\sin(\dfrac{-\pi/2+2\pi(0)}{3}))

=32i2=\dfrac{\sqrt{3}}{2}-\dfrac{i}{2}

k=1:k=1:


13(cos(π/2+2π(1)3)+isin(π/2+2π(1)3))\sqrt[3]{1}(\cos(\dfrac{-\pi/2+2\pi(1)}{3})+i\sin(\dfrac{-\pi/2+2\pi(1)}{3}))

=i=i

k=2:k=2:


13(cos(π/2+2π(2)3)+isin(π/2+2π(2)3))\sqrt[3]{1}(\cos(\dfrac{-\pi/2+2\pi(2)}{3})+i\sin(\dfrac{-\pi/2+2\pi(2)}{3}))

=32i2=-\dfrac{\sqrt{3}}{2}-\dfrac{i}{2}


z1=3i,z2=2i,z3=3i,z_1=\sqrt{3}-i, z_2=2i,z_3=-\sqrt{3}-i,


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