Answer to Question #349058 in Complex Analysis for Mapula Advice

Question #349058

Which of the following is a solution of 2z3+16i=0 ?


1
Expert's answer
2022-06-13T17:49:40-0400
"2z^3+16i=0"

"z^3=-8i"

The polar form of "- 8 i"  is "8(\\cos(-\\dfrac{\\pi}{2})+i\\sin(-\\dfrac{\\pi}{2}))."

"k=0:"


"\\sqrt[3]{8}(\\cos(\\dfrac{-\\pi\/2+2\\pi(0)}{3})+i\\sin(\\dfrac{-\\pi\/2+2\\pi(0)}{3}))"

"=\\sqrt{3}-i"

"k=1:"


"\\sqrt[3]{8}(\\cos(\\dfrac{-\\pi\/2+2\\pi(1)}{3})+i\\sin(\\dfrac{-\\pi\/2+2\\pi(1)}{3}))"

"=i"

"k=2:"


"\\sqrt[3]{8}(\\cos(\\dfrac{-\\pi\/2+2\\pi(2)}{3})+i\\sin(\\dfrac{-\\pi\/2+2\\pi(2)}{3}))"

"=-\\sqrt{3}-i"

The solutions are "\\{-\\sqrt{3}-i, i, \\sqrt{3}-i\\}."



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