Find all solutions to Z3 8iSolutionZ38i Za bi abi38i a3b i

Find all solutions to Z^3 = -8i

Solution

Z³=-8i

Z=a + bi

(a+bi)³=-8i

a³+(b i)³ + 3a²b i +   3a (b i)²=-8i

a³ -   b³i + 3a²b i - 3ab²=-8i

a³-3ab² - i(b³-3a²b) =-8i

a³-3ab²=0

b³-3a²b=8

From equation a³-3ab=0

we get

a(a²-3b²)=0

a=0 or a²=3b²

when a=0, b³-3(0)²b=8

                b³=8

b=2

when a²=3b²

b³-3(3b²)b=8

b³-9b³=8

-8b³=8

b=-1

and a²=3b²

a²=3

a=+-sqrt3

So when a=0,b=2

Therefore we have a+bi = 0+2i=2i

And when a=+-sqrt3,b=-1

a +bi

sqrt3 - 1i    , -sqrt3 -1i