Question

Qual é o inverso multiplicativo do número complexo z = 3 – 2i.
13 pontos

A- 3/5 + 2i/5
B- 3/10 + 1i/5
C- 3/13 - 2i/13
D- 3/13 + 2i/13

Answer 1

1/3-2i * (3+2i)/(3+2i)
3+2i/9+6i-6i-4i²  > i² = -1  então -4*-1 = 4
3+2i/9+4

3/13 + 2i+13
Alternativa D 

Answer 2

✅ Após resolver os cálculos, concluímos que o inverso do referido número complexo é:

$\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf I(z) = \frac{3}{13} + \frac{2}{13}i \:\:\:}}\end{gathered} }$

Portanto, a opção correta é:

       $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf Letra\:D\:\:\:}}\end{gathered} }$

Seja o número complexo:

         $\Large\displaystyle\begin{gathered} z= 3 - 2i\end{gathered}$

Sabendo que o inverso multiplicativo ou recíproco de um número pode ser representado por:

       $\Large\displaystyle\begin{gathered} I(n) = \frac{1}{n} \end{gathered}$

Se:

            $\Large\displaystyle\begin{gathered} n = z\end{gathered}$

Então, temos:

     $\Large\displaystyle\begin{gathered} I(z) = \frac{1}{z} \end{gathered}$

                $\Large\displaystyle\begin{gathered} = \frac{1}{3 - 2i} \end{gathered}$

                $\Large\displaystyle\begin{gathered} = \frac{1}{3 - 2i} \cdot\frac{3 + 2i}{3 + 2i} \end{gathered}$

                $\Large\displaystyle\text{ \begin{gathered} = \frac{3 + 2i}{9 - 6i + 6i - 4i^{2}} \end{gathered} }$

                $\Large\displaystyle\text{ \begin{gathered} = \frac{3 + 2i}{9 - 4i^{2}} \end{gathered} }$

                $\Large\displaystyle\begin{gathered} = \frac{3 + 2i}{9 - 4\cdot(-1)} \end{gathered}$

                $\Large\displaystyle\begin{gathered} = \frac{3 + 2i}{9 + 4} \end{gathered}$

                $\Large\displaystyle\begin{gathered} = \frac{3 + 2i}{13} \end{gathered}$

                $\Large\displaystyle\begin{gathered} = \frac{3}{13} + \frac{2}{13}i \end{gathered}$

✅ Portanto, o inverso do referido número é:

      $\Large\displaystyle\begin{gathered} I(z) =\frac{3}{13} + \frac{2}{13}i \end{gathered}$

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