What are the solutions to the equation z^3-8i=0 ?

The solutions to the equation z^3-8i=0 are z=sqrt(3)+i,z=-sqrt(3)+i,z=-2i

Find the zeros of z^3-8i=0

The zeros of z^3-8i=0 are z=sqrt(3)+i,z=-sqrt(3)+i,z=-2i

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z^{3} - 8 i = 0

z = 3 + i, z = - 3 + i, z = - 2 i

Solution steps

z^{3} - 8 i = 0

Move

8 i

to the right side

z^{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}

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

Simplify

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

z = 3 + i, z = - 3 + i, z = - 2 i

3 x^{3} + 7 x^{2} + 6 x - 8 = 0

x^{4} + 55 x^{2} - 576

y^{3} - 9 y^{2} + 8 y + 60 = 0

2 x^{5} - 18 x = 0

x^{9} - x^{6} = 36

What are the solutions to the equation z^3-8i=0 ?
The solutions to the equation z^3-8i=0 are z=sqrt(3)+i,z=-sqrt(3)+i,z=-2i
Find the zeros of z^3-8i=0
The zeros of z^3-8i=0 are z=sqrt(3)+i,z=-sqrt(3)+i,z=-2i