Question

Calcule o modulo de z = (1 + i / 1 - 2i)²?

Answer 1

z= ( 1+ i ) ² / ( 1- 2i ) 

z= ( 1+2i-1 ) / ( 1- 2i ) 

z= 2i / ( 1- 2i ) 

z= 2i(1+2i) / ( 1- 2i )(1+2i) 

z=(2i-4)/(1+4) 

z=2i/5 -4/5 

|z|=[16/25+4/25]¹/² 

|z|=[4/5]¹/²=2/√5=2√5/5

Answer 2

$Z=(\frac{1+i}{1-2i})^2\\\\Z=\frac{1+i}{1-2i}*\frac{1+i}{1-2i}\\\\Z=\frac{1+2i+i^2}{1-4i+4i^2}\\\\Z=\frac{1+2i+(-1)}{1-4i+4(-1)}\\\\Z=\frac{2i}{1-4i-4}\\\\Z=\frac{2i}{-3-4i}\\\\Z=\frac{2i}{-3-4i}*\frac{-3+4i}{-3+4i}\\\\Z=\frac{-6i+6i^2}{9-16i^2}\\\\Z=\frac{-6i+6(-1)}{9-16(-1)}\\\\Z=\frac{-6i-6}{9+16}\\\\Z=\frac{-6i-6}{25}\\\\Z=\frac{-6i}{25}-\frac{6}{25}\\\\Z=-0,24i-0,24\\\\\\\\|Z|=|-0,24-0,24i|\\\\Z=0,24+0,24i$