Escreva na forma $a + bi$
Números complexos: Infinito 12 A - Parte 3 Pág. 72 Ex. 39
Enunciado
Escreva na forma $a + bi$:
- $\frac{5}{{3 – i}}$
- $\frac{{2 + i}}{{2 – i}}$
- $\frac{{3 + 2i}}{{5i}}$
- ${i^{101}}$
- ${i^{1999}} – 2$
- ${i^{4n}} – 2{i^{4n + 3}}$
Resolução
- Ora,
$$\begin{array}{*{20}{l}}
{\frac{5}{{3 – i}}}& = &{\frac{5}{{3 – i}} \times \frac{{3 + i}}{{3 + i}}} \\
{}& = &{\frac{{15 + 5i}}{{{3^2} + 1}}} \\
{}& = &{\frac{3}{2} + \frac{1}{2}i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{\frac{{2 + i}}{{2 – i}}}& = &{\frac{{2 + i}}{{2 – i}} \times \frac{{2 + i}}{{2 + i}}} \\
{}& = &{\frac{{4 + 2i + 2i – 1}}{{4 + 1}}} \\
{}& = &{\frac{3}{5} + \frac{4}{5}i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{\frac{{3 + 2i}}{{5i}}}& = &{\frac{{3 + 2i}}{{5i}} \times \frac{{ – i}}{{ – i}}} \\
{}& = &{\frac{{2 – 3i}}{5}} \\
{}& = &{\frac{2}{5} – \frac{3}{5}i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{{i^{101}}}& = &{{{\left( {{i^4}} \right)}^{25}} \times i} \\
{}& = &{{1^{25}} \times i} \\
{}& = &i
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{{i^{1999}} – 2}& = &{{{\left( {{i^4}} \right)}^{499}} \times {i^3} – 2} \\
{}& = &{1 \times \left( { – i} \right) – 2} \\
{}& = &{ – 2 – i}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{{i^{4n}} – 2{i^{4n + 3}}}& = &{{{\left( {{i^4}} \right)}^n} – 2 \times {{\left( {{i^4}} \right)}^n} \times {i^3}} \\
{}& = &{1 – 2 \times 1 \times \left( { – i} \right)} \\
{}& = &{1 + 2i}
\end{array}$$
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