Question

Determine o módulo do complexo 2+3i/-5-i

Answer 1

Se $z=\dfrac{z_{1}}{z_{2}}$, então $|z|=\bigg|\dfrac{z_{1}}{z_{2}}\bigg|=\dfrac{|z_{1}|}{|z_{2}|}$
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$z=\dfrac{2+3i}{-5-i}$

Sendo $z_{1}=2+3i$ e $z_{2}=-5-i$, então

$z=\dfrac{z_{1}}{z_{2}}~~\Longrightarrow~~|z|=\dfrac{|z_{1}|}{|z_{2}|}$

Vamos achar o módulo de $z_{1}$ e $z_{2}$:

$\bullet\,\,\,z_{1}=2+3i~~\Longrightarrow~~Re(z_{1})=2~~\mathsf{e}~~Im(z_{1})=3\\\\|z_{1}|=\sqrt{\big[Re(z_{1})\big]^{2}+\big[Im(z_{1})\big]^{2}}=\sqrt{2^{2}+3^{2}}=\sqrt{4+9}=\sqrt{13}\\\\\\\bullet\,\,\,z_{2}=-5-i=-5-1i~~\Longrightarrow~~Re(z_{2})=-5~~\mathsf{e}~~Im(z_{2})=-1\\\\|z_{2}|=\sqrt{\big[Re(z_{2})\big]^{2}+\big[Im(z_{2})\big]^{2}}=\sqrt{(-5)^{2}+(-1)^{2}}=\sqrt{25+1}=\sqrt{26}$

Logo,

$|z|=\dfrac{|z_{1}|}{|z_{2}|}\\\\\\|z|=\dfrac{\sqrt{13}}{\sqrt{26}}\\\\\\|z|=\dfrac{\sqrt{13}}{\sqrt{2\cdot13}}\\\\\\|z|=\dfrac{\sqrt{13}}{\sqrt{2}\cdot\sqrt{13}}\\\\\\|z|=\dfrac{1}{\sqrt{2}}\\\\\\|z|=\dfrac{\sqrt{2}}{\sqrt{2}\cdot\sqrt{2}}\\\\\\\boxed{\boxed{|z|=\dfrac{\sqrt{2}}{\,2}}}$