Question

Calcule o módulo do número complexo​

Answer 1

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$\Large\boxed{\sf{\underline{N\acute{u}mero~complexo~na~forma~alg\acute{e}brica}}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf{z=a+bi}}}}}$

$\Large\boxed{\sf{\underline{M\acute{o}dulo~de~um~n\acute{u}mero~complexo}}}}\\\huge\boxed{\boxed{\boxed{\boxed{\boxed{\sf{\rho=\sqrt{a^2+b^2}}}}}}}$

$\sf{z=\dfrac{1}{2}+\dfrac{1}{3}i}\\\sf{\rho=\sqrt{\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{3}\right)^2}}\\\sf{\rho=\sqrt{\dfrac{1}{4}+\dfrac{1}{9}}}\\\sf{\rho=\sqrt{\dfrac{9+4}{36}}}\\\sf{\rho=\sqrt{\dfrac{13}{36}}}$

$\sf{\rho=\dfrac{\sqrt{13}}{\sqrt{36}}$

$\huge\boxed{\boxed{\boxed{\boxed{\sf{\rho=\dfrac{\sqrt{13}}{6}}}}}}$

Answer 2

✅ Tendo terminado 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:

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

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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