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heureka

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 #2
avatar+26396 
+5

find all the fourth roots of -8 in Cartesian form.

simplify as much as possilbe

 

z4=8z1, z2, z3, z4= ?z4=(2)4|±z2=±22z2=±2(1)2z2=±221|1=iz2=±22i

 

Solutions z1, z2, z3, z4:

(1)z21=+22i(2)z22=(z1)2=+22iorz2=z1(3)z23=22i(4)z24=(z3)2=22iorz4=z3

 

z1(1)z21=+22i|z1=a+bi(a+bi)2=+22ia2+2abi+b2i2=+22i|i2=1a2+2abib2=+22ia2b2Re=0+2abiIm=+22i=+22ia2b2=0a2=b22abi=+22iab=2b=2aa2=b2a2=(2a)2a2=2a2a4=2a=42b=2ab=242b=212214b=21214b=214b=42z1=a+biz1=42+42iz1=42(1+i)

 

z2(2)z2=z1z2=42(1+i)

 

z3(3)z23=22i|z3=a+bi(a+bi)2=22ia2+2abi+b2i2=22i|i2=1a2+2abib2=22ia2b2Re=0+2abiIm=22i=22ia2b2=0a2=b22abi=22iab=2b=2aa2=b2a2=(2a)2a2=2a2a4=2a=42b=2ab=242b=212214b=21214b=214b=42z3=a+biz3=4242iz3=42(1i)

 

z4(4)z4=z3z4=42(1i)

 

z1=42(1+i)z2=42(1+i)z3=42(1i)z4=42(1i)

 

laugh

20.10.2016
 #4
avatar+26396 
+5
19.10.2016