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Question Number 139867 by Maclaurin Stickker last updated on 01/May/21

a^{3} + b^{3} + c^{3} = 0

a^{12} + b^{12} + c^{12} = 8

a^{6} + b^{6} + c^{6} = ?

(a n s : 4)

Answered by MJS_new last updated on 02/May/21

x + y + z = 0

x^{4} + y^{4} + z^{4} = 8

x^{2} + y^{2} + z^{2} = ?

z = - (x + y)

x^{4} + y^{4} + {(x + y)}^{4} = 8

x = u - v \land y = u + v

{(u - v)}^{4} + {(u + v)}^{4} + {(2 u)}^{4} = 8

9 u^{4} + 6 u^{2} v^{2} + v^{4} = 4

x^{2} + y^{2} + {(x + y)}^{2} =

= {(u - v)}^{2} + {(u + v)}^{2} + {(2 u)}^{2} = 6 u^{2} + 2 v^{2}

{(6 u^{2} + 2 v^{2})}^{2} = 4 {(3 u^{2} + v^{2})}^{2} = 2 (9 u^{4} + 6 u^{2} v^{2} + v^{4}) =

= 4 \times 4 = 16 \Rightarrow 6 u^{2} + 2 v^{2} = 4 \Rightarrow \dots x^{2} + y^{2} + z^{2} = 4 \Rightarrow

\Rightarrow a^{6} + b^{6} + c^{6} = 4

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