What are the solutions to the equation z^6-2z^3+2=0 ?

The solutions to the equation z^6-2z^3+2=0 are

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z^6-2z^3+2=0

z^{6} - 2 z^{3} + 2 = 0

z = \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}}}{2} + i \frac{2 ^{\frac{2}{3}}}{2}, z = \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 - 3 )}{4}, z = \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}}}{2} - i \frac{2 ^{\frac{2}{3}}}{2}

Solution steps

z^{6} - 2 z^{3} + 2 = 0

Rewrite the equation with

u = z^{3}

and

u^{2} = z^{6}

u^{2} - 2 u + 2 = 0

Solve

u^{2} - 2 u + 2 = 0 : u = 1 + i, u = 1 - i

u = 1 + i, u = 1 - i

Substitute back

u = z^{3},

solve for

z

Solve

z^{3} = 1 + i : z = \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}}}{2} + i \frac{2 ^{\frac{2}{3}}}{2}, z = \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 - 3 )}{4}

Solve

z^{3} = 1 - i : z = \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}}}{2} - i \frac{2 ^{\frac{2}{3}}}{2}

The solutions are

z = \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}}}{2} + i \frac{2 ^{\frac{2}{3}}}{2}, z = \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( - 1 - 3 )}{4}, z = \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 - 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}} ( - 1 + 3 )}{4} + i \frac{2 ^{\frac{2}{3}} ( 1 + 3 )}{4}, z = - \frac{2 ^{\frac{2}{3}}}{2} - i \frac{2 ^{\frac{2}{3}}}{2}

z^{10} = 9^{10}

(x^{2} - 7)^{2} - 5 (x^{2} - 7) + 6 = 0

x^{3} - 3 x^{2} + 3 x - 1

x^{4} = 12

y, x^{3} - 4 x y^{2} - 16 = 3 x - y^{4}

What are the solutions to the equation z^6-2z^3+2=0 ?
The solutions to the equation z^6-2z^3+2=0 are
Find the zeros of z^6-2z^3+2=0
The zeros of z^6-2z^3+2=0 are