SOLUTION: hello mam\sir i have some doubts in following questions 1.find the cube root of 3+isquare root 3 2.if z=x+iy ,proove that |x|+|y|less than or equal to sqareroot2|z| 3.proove [

Question 12254: hello mam\sir i have some doubts in following questions
1.find the cube root of 3+isquare root 3
2.if z=x+iy ,proove that |x|+|y|less than or equal to sqareroot2|z|
3.proove [ a+b omega+c(omega)^2] divided by [a omega+b (omega)^2+c]=(omega)^2
4.proove 1+(omega)^n+(omega)^2n=0 when n=2,4
thank you in advance
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
1.find the cube root of 3+isquare root 3
Do you mean cubic root of(3 + i sqrt(3)) I guess do.
To solve:
Since
where r = sqrt(3^2+ 3) =
, i.e. w = pi/6.
By De Mieve theorem, ,k = 0,1,2
Hence,

2.if z=x+iy ,prove that |x|+|y| less than or equal to sqareroot(2)|z|
proof: Since
and
Consider
=
=
Hence,
By taking sqrt, we have

**You should define what w (omega) is, I guess, it is

3.prove [ a+b omega+c(omega)^2] divided by [a omega+b (omega)^2+c]=(omega)^2
Proof:
=
=
[ Since , ]
=
= [ is cancelled ]
4.prove 1+(omega)^n+(omega)^2n=0 when n=2,4
Proof: Since
=
=
and
=
=
When n =2,
[Note: ]
=
=
{or use ew is the primitive root of directly]

Similarly, when n =4,
[Note: ]

Kenny
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