What are the solutions to the equation x^3=-8i ?

The solutions to the equation x^3=-8i are x=sqrt(3)-i,x=2i,x=-sqrt(3)-i

Find the zeros of x^3=-8i

The zeros of x^3=-8i are x=sqrt(3)-i,x=2i,x=-sqrt(3)-i

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x^{3} = - 8 i

x = 3 - i, x = 2 i, x = - 3 - i

Solution steps

x^{3} = - 8 i

For

z^{n} = a

the solutions are

z_{k} = n ∣ a ∣ (cos (\frac{ar g ( a ) + 2 kπ}{n}) + i sin (\frac{ar g ( a ) + 2 kπ}{n})),

k = 0, 1, \dots, n - 1

For

n = 3, a = - 8 i

∣ a ∣ = 8

ar g (a) = - \frac{π}{2}

x = 38 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{3})), x = 38 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{3})), x = 38 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{3}))

Simplify

38 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{3})) : 3 - i

Simplify

38 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{3})) : 2 i

Simplify

38 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{3})) : - 3 - i

x = 3 - i, x = 2 i, x = - 3 - i

2 x^{3} - 10 x^{2} - 72 x = 0

x + (2 x^{2} - 3)^{3} = 0

- 4 x^{3} + 16 x = 0

x^{4} - 5 x^{2} - 36

x^{3} - 81 = 0

What are the solutions to the equation x^3=-8i ?
The solutions to the equation x^3=-8i are x=sqrt(3)-i,x=2i,x=-sqrt(3)-i
Find the zeros of x^3=-8i
The zeros of x^3=-8i are x=sqrt(3)-i,x=2i,x=-sqrt(3)-i