Question

calcule o módulo dos complexos abaixo:

a) z=√3 + i / √3 - i

b)√2 + √10 i

​

Answer 1

Resposta:

a) |z| = 1

b) |z| = 2√3

Explicação passo a passo:

$a)\ z=\frac{\sqrt{3} +i}{\sqrt{3} -i} \\\\\\z=\frac{(\sqrt{3} +i)(\sqrt{3} +i)}{(\sqrt{3} -i)(\sqrt{3} +i)} \\\\\\z=\frac{(\sqrt{3} +i)^{2} }{(\sqrt{3})^{2} -i^{2} } \\\\\\z=\frac{(\sqrt{3} )^{2}+2i\sqrt{3}+i^{2} }{3 -(-1) } \\\\\\z=\frac{3+2i\sqrt{3}+(-1) }{3+1 } \\\\\\z=\frac{2+2i\sqrt{3} }{4 } \\\\\\z=\frac{1+i\sqrt{3} }{2 } \\\\\\z=\frac{1}{2} +\frac{\sqrt{3} }{2 } i$

$|z|=\sqrt{(\frac{1}{2} )^{2} +(\frac{\sqrt{3} }{2} )^{2} } \\\\\\|z|=\sqrt{\frac{1^{2} }{2^{2} } +\frac{(\sqrt{3})^{2} }{2^{2} } } \\\\\\|z|=\sqrt{\frac{1}{4} +\frac{3}{4} } \\\\\\|z|=\sqrt{\frac{1+3}{4} } \\\\\\|z|=\sqrt{\frac{4}{4} } \\\\\\|z|=\sqrt{1 } \\\\\\\bold{|z|=1}$

$------------------------$

$b)\ z= \sqrt{2} +\sqrt{10} i\\\\|z|=\sqrt{(\sqrt{2})^{2}+(\sqrt{10} )^{2} } \\\\|z|=\sqrt{2+10 } \\\\|z|=\sqrt{12 }\\\\|z|=\sqrt{4.3 }\\\\|z|=\sqrt{4} \sqrt{3} \\\\\bold{|z|=2\sqrt{3} }$