Question

Calcule o módulo do número complexo​

Answer 1

Resposta:

$|z| = \frac{ \sqrt{13} }{6}$

Explicação passo-a-passo:

$z = \frac{1}{2} + \frac{1}{3} i$

$|z| = | \frac{1}{2} + \frac{1}{3} i|$

$|z| = \sqrt{ {( \frac{1}{2} )}^{2} + {( \frac{1}{3}) }^{2} }$

$|z| = \sqrt{ \frac{1}{4} + \frac{1}{9} }$

$|z| = \sqrt{ \frac{13}{36} }$

$|z| = \frac{ \sqrt{13} }{6}$

Answer 2

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

     $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf \parallel z \parallel = \frac{\sqrt{13}}{6}\:\:\:}}\end{gathered} }$

 

Seja o número complexo:

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

Uma vez sabendo que todo número complexo em sua forma algébrica pode ser montada da seguinte forma:

         $\Large\displaystyle\begin{gathered} z = a + bi\end{gathered}$

Desta forma, podemos calcular o módulo do número complexo fazendo:

       $\Large\displaystyle\text{ \begin{gathered} z = \sqrt{a^{2} + b^{2}}\end{gathered} }$

Então:

  $\Large\displaystyle\text{ \begin{gathered} \parallel z \parallel= \sqrt{\bigg(\frac{1}{2}\bigg)^{2} + \bigg(\frac{1}{3}\bigg)^{2}}\end{gathered} }$

            $\Large\displaystyle\text{ \begin{gathered} = \sqrt{\frac{1^{2}}{2^{2}} + \frac{1^{2}}{3^{2}}}\end{gathered} }$

            $\Large\displaystyle\text{ \begin{gathered} = \sqrt{\frac{1}{4} + \frac{1}{9}}\end{gathered} }$

            $\Large\displaystyle\text{ \begin{gathered} = \sqrt{\frac{9 + 4}{36}}\end{gathered} }$

            $\Large\displaystyle\text{ \begin{gathered} = \sqrt{\frac{13}{36}}\end{gathered} }$

            $\Large\displaystyle\text{ \begin{gathered} = \frac{\sqrt{13}}{\sqrt{36}}\end{gathered} }$

             $\Large\displaystyle\text{ \begin{gathered} = \frac{\sqrt{13}}{6}\end{gathered} }$

✅ Portanto, o módulo procurado é:

      $\Large\displaystyle\text{ \begin{gathered} \parallel z \parallel = \frac{\sqrt{13}}{6}\end{gathered} }$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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