Quais são as raízes da equação X ao cubo-8=0
hcsmalves:
Em qual conjunto R ou C?
Se for em R a raiz é 2.
Ele disse que nessa equação haveria 3 raízes
Soluções para a tarefa
Respondido por
1
x³ - 8 = 0 => x³ = 8 => x = ∛8
Seja Z = 8 => |Z| = 8
cosα = a/|Z| => cosα = 8/8 = 1
senα = b/|Z| => senα = 0
Logo, α = 0
![Z_{k} = \sqrt[n]{|Z|}(cos \frac{ \alpha +2k \pi }{n} +isen \frac{ \alpha +2k \pi }{n} ) \\ \\ p/k=0=\ \textgreater \ Z_{0} = \sqrt[3]{|8|} (cos \frac{0+2.0. \pi }{3} +isen \frac{0+2.0. \pi }{3} )=\ \textgreater \ \\ \\ k_{0}=2(cos0+isen0)=\ \textgreater \ k_{0}=2(1+i.0)=\ \textgreater \ k_{0}=2 \\ \\ p/k=1=\ \textgreater \ k_{1} =2(cos \frac{0+2.1. \pi }{3}+isen \frac{0+2.1. \pi }{3}) \\ \\ k_{1} =2(cos \frac{2 \pi }{3} + \frac{2 \pi }{3)} =\ \textgreater \ k_{1}=2(- \frac{1}{2} +i. \frac{ \sqrt{3} }{2} )=\ \textgreater \ k_{1}=-1+ \sqrt{3}i \\ \\](https://tex.z-dn.net/?f=+Z_%7Bk%7D+%3D+%5Csqrt%5Bn%5D%7B%7CZ%7C%7D%28cos+%5Cfrac%7B+%5Calpha+%2B2k+%5Cpi+%7D%7Bn%7D+%2Bisen+%5Cfrac%7B+%5Calpha+%2B2k+%5Cpi+%7D%7Bn%7D+%29+%5C%5C++%5C%5C+p%2Fk%3D0%3D%5C+%5Ctextgreater+%5C++Z_%7B0%7D+%3D+%5Csqrt%5B3%5D%7B%7C8%7C%7D+%28cos+%5Cfrac%7B0%2B2.0.+%5Cpi+%7D%7B3%7D+%2Bisen+%5Cfrac%7B0%2B2.0.+%5Cpi+%7D%7B3%7D+%29%3D%5C+%5Ctextgreater+%5C+++%5C%5C++%5C%5C++k_%7B0%7D%3D2%28cos0%2Bisen0%29%3D%5C+%5Ctextgreater+%5C++k_%7B0%7D%3D2%281%2Bi.0%29%3D%5C+%5Ctextgreater+%5C++k_%7B0%7D%3D2+%5C%5C++%5C%5C+p%2Fk%3D1%3D%5C+%5Ctextgreater+%5C++k_%7B1%7D+%3D2%28cos+%5Cfrac%7B0%2B2.1.+%5Cpi+%7D%7B3%7D%2Bisen+%5Cfrac%7B0%2B2.1.+%5Cpi+%7D%7B3%7D%29+%5C%5C++%5C%5C++k_%7B1%7D+%3D2%28cos+%5Cfrac%7B2+%5Cpi+%7D%7B3%7D+%2B+%5Cfrac%7B2+%5Cpi+%7D%7B3%29%7D++%3D%5C+%5Ctextgreater+%5C++k_%7B1%7D%3D2%28-+%5Cfrac%7B1%7D%7B2%7D++%2Bi.+%5Cfrac%7B+%5Csqrt%7B3%7D+%7D%7B2%7D+%29%3D%5C+%5Ctextgreater+%5C++k_%7B1%7D%3D-1%2B+%5Csqrt%7B3%7Di++%5C%5C++%5C%5C+++++++++ "Z_{k} = \sqrt[n]{|Z|}(cos \frac{ \alpha +2k \pi }{n} +isen \frac{ \alpha +2k \pi }{n} ) \\ \\ p/k=0=\ \textgreater \ Z_{0} = \sqrt[3]{|8|} (cos \frac{0+2.0. \pi }{3} +isen \frac{0+2.0. \pi }{3} )=\ \textgreater \ \\ \\ k_{0}=2(cos0+isen0)=\ \textgreater \ k_{0}=2(1+i.0)=\ \textgreater \ k_{0}=2 \\ \\ p/k=1=\ \textgreater \ k_{1} =2(cos \frac{0+2.1. \pi }{3}+isen \frac{0+2.1. \pi }{3}) \\ \\ k_{1} =2(cos \frac{2 \pi }{3} + \frac{2 \pi }{3)} =\ \textgreater \ k_{1}=2(- \frac{1}{2} +i. \frac{ \sqrt{3} }{2} )=\ \textgreater \ k_{1}=-1+ \sqrt{3}i \\ \\")
p/k=2=>
S = {2, -1-√3i, -1+√3i}
Seja Z = 8 => |Z| = 8
cosα = a/|Z| => cosα = 8/8 = 1
senα = b/|Z| => senα = 0
Logo, α = 0
p/k=2=>
S = {2, -1-√3i, -1+√3i}
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