Determine
Números complexos: Infinito 12 A - Parte 3 Pág. 102 Ex. 62
Enunciado
Determine:
- as cinco raízes quintas de $z = 1$;
- as quatro raízes quartas de $z = i$.
Resolução
- As cinco raízes quintas de $z = 1 = \operatorname{cis} \left( 0 \right)$ são:
$$\begin{array}{*{20}{l}}
{k = 0:}&{{w_0} = \sqrt[5]{1}\operatorname{cis} \left( {\frac{0}{5}} \right) = \operatorname{cis} \left( 0 \right)} \\
{k = 1:}&{{w_1} = \sqrt[5]{1}\operatorname{cis} \left( {\frac{0}{5} + \frac{{2\pi }}{5}} \right) = \operatorname{cis} \left( {\frac{{2\pi }}{5}} \right)} \\
{k = 2:}&{{w_2} = \sqrt[5]{1}\operatorname{cis} \left( {\frac{0}{5} + \frac{{4\pi }}{5}} \right) = \operatorname{cis} \left( {\frac{{4\pi }}{5}} \right)} \\
{k = 3:}&{{w_3} = \sqrt[5]{1}\operatorname{cis} \left( {\frac{0}{5} + \frac{{6\pi }}{5}} \right) = \operatorname{cis} \left( {\frac{{6\pi }}{5}} \right)} \\
{k = 4}&{{w_4} = \sqrt[5]{1}\operatorname{cis} \left( {\frac{0}{5} + \frac{{8\pi }}{5}} \right) = \operatorname{cis} \left( {\frac{{8\pi }}{5}} \right)}
\end{array}$$
- As quatros raízes quartas de $z = i = \operatorname{cis} \left( {\frac{\pi }{2}} \right)$ são:
$$\begin{array}{*{20}{l}}
{k = 0:}&{{w_0} = \sqrt[4]{1}\operatorname{cis} \left( {\frac{{\tfrac{\pi }{2}}}{4}} \right) = \operatorname{cis} \left( {\frac{\pi }{8}} \right)} \\
{k = 1:}&{{w_1} = \sqrt[4]{1}\operatorname{cis} \left( {\frac{{\tfrac{\pi }{2}}}{4} + \frac{{2\pi }}{4}} \right) = \operatorname{cis} \left( {\frac{{5\pi }}{8}} \right)} \\
{k = 2:}&{{w_2} = \sqrt[4]{1}\operatorname{cis} \left( {\frac{{\tfrac{\pi }{2}}}{4} + \frac{{4\pi }}{4}} \right) = \operatorname{cis} \left( {\frac{{9\pi }}{8}} \right)} \\
{k = 3:}&{{w_3} = \sqrt[4]{1}\operatorname{cis} \left( {\frac{{\tfrac{\pi }{2}}}{4} + \frac{{6\pi }}{4}} \right) = \operatorname{cis} \left( {\frac{{13\pi }}{8}} \right)}
\end{array}$$
