Carla

a) -5 + 12i

raiz( -5 + 12i )= x+ iy

(-5+12i)=(x+yi)^2= x^2+2xy +(yi)^2

mas i^2=-1

assim

(-5+12i)=x^2-y^2+2xyi

x^2-y^2==5

2xy=12~~>y=(6/x)

x^2-(6/x)^2 = -5

x^2-36/x^2=-5

x^4+5x^2-36=0

fazendo z=x^2

z^2+5z-36-0

z=(-5+/-raiz(5^2+4*36))/2

z=(-5+/-13)/2

z1=-9 descarte por enquanto

z2=4= x^2

x=+/-2

para x=2~~y=6/x= 6/2=3

~~~~>raiz(-5+12i)=2+3i

para x= -2~~> y=6/-2=-3

~~~~>raiz(-5+12i)=-2-3i

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b) 4i

raiz( 4i )=x+iy

4i= x^2-y^2+2xyi

x^2-y^2=0

2xy=4~~> y=2/x

x^2-(2/x)^2=0

x^2-4/x^2=0

x^4-4=0~~>x^4=4~~~> x^2=2

x= +/-raiz(2)

para x= raiz(2)

y=2/x= 2/raiz(2)=raiz(2)/2

raiz(4i) = raiz(2) +i* raiz(2)/2

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c) 4 + 3i

raiz( 4 + 3i)= x+ iy

4 + 3i = x^2-y^2+2xyi

x^2-y^2=4

2xy=3~~> y=3/(2x)

x^2-(3/(2x))^2=4

x^2-9/(4x^2)=4

4x^4-16x^2-9=0

para z=x^2

4z^2-16z-9=0

z= (16+/- raiz(16^2+4*4*9))/8

z=(16+/-raiz(400))/8

z=(16+/-20)/8= (9/2) ~~descarta a raiz negativa

z=9/2= x^2

x= +/- 3/raiz(2)

para x= 3/raiz(2)

y=3/2x=3/2(3/raiz(2))=(3/2)*raiz(2)/3= raiz(2)/2

~~~>raiz(4 + 3i )= (3/raiz(2))+ i*raiz(2)/2

igualmenrte para x= -3/raiz(2)

~~~>raiz(4 + 3i )= (-3/raiz(2))- i*raiz(2)/2

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d) 1 - i V3

raiz( 1 - i V3)= x+iy

1 - i V3= x^2-y^2+ 2xyi

x^2-y^2=1

2xy=-raiz(3)

y= -raiz(3)/2x

x^2-3/(4x^2) =1

4x^4-4x^2-3=0

para z= x^2

4z^2-4z-3=0

z=(4+/-raiz(4^2+4*4*3))/8

z=(4+/- 8)/8

z= 12/8=(3/2)~~> raiz negativa regeitada

z=x^2= 3/2

x = +/- raiz(3/2)

para x= raiz(3)/raiz(2)

y=-raiz(3)/(2*( raiz(3)/raiz(2))=

y=-raiz(2)/2

~~~>raiz(1 - i V3)=raiz(3/2) -i*raiz(2)/2

para x= -raiz(3/2) ~~~> y= raiz(2)/2

e ~~~>raiz(1 - i V3)=-raiz(3/2) +i*raiz(2)/2

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