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 [

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> 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 [      Log On


   



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) About Me  (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: +x%5E3+=+3+%2B+i+sqrt%283%29
Since +3+%2B+i+sqrt%283%29+=+r++%28cos+w++%2B+i+sin+w%29+
where r = sqrt(3^2+ 3) = 2+sqrt%283%29+
+tan+w+=+sqrt%283%29%2F3+=+1%2Fsqrt%283%29+, i.e. w = pi/6.
By De Mieve theorem, ,k = 0,1,2
Hence,
+primitive+cubic+root%2C+pi%2F18+=+20+deg+
or
+13%2A+pi%2F18+=+130+deg+
or
+25%2A+pi%2F18+=+250+deg+
2.if z=x+iy ,prove that |x|+|y| less than or equal to sqareroot(2)|z|
proof: Since +%28sqrt%282%29%2Aabs%28z%29%29%5E2+=+2%2Aabs%28z%29%5E2+=+2%2A%28x%5E2+%2B+y%5E2%29+
and +%28abs%28x%29+%2B+abs%28y%29%29%5E2+=+x%5E2+%2B+y%5E2++%2B+2+%2Aabs%28x%29+abs%28y%29+
Consider +2%2A%28x%5E2+%2B+y%5E2%29+-+%28x%5E2+%2B+y%5E2++%2B+2%2Aabs%28x%29+abs%28y%29%29++
= ++x%5E2+%2B+y%5E2++-+2%2Aabs%28x%29%2Aabs%28y%29+
= +%28abs%28x%29+-abs%28y%29%29%5E2+%3E=+0+
Hence, +%28+abs%28x%29+%2B+abs%28y%29%29%5E2+%3C=+2%2A+%28abs%28z%29%29%5E2++
By taking sqrt, we have +abs%28x%29+%2B+abs%28y%29+%3C=+sqrt%282%29%2Aabs%28z%29+

**You should define what w (omega) is, I guess, it is
+e%5E%282%2Api%2Ai%2F3%29+=+cos+%282%2Api%2F3%29+%2B+i+sin+%282%2Api%2F3%29+
3.prove [ a+b omega+c(omega)^2] divided by [a omega+b (omega)^2+c]=(omega)^2
Proof: %28a+%2B+b+%2Aw++%2B+c+%2Aw%5E2%29%2F+%28a+%2Aw+%2B+b+%2Aw%5E2++%2B+c%29+
=
=
[ Since +w%5E3+=+1+, w%5E4+=+w+ ]
= +w%5E2+%2A%28a+%2B+b+%2Aw++%2B+c+%2Aw%5E2%29%2F+%28a+%2B+b+%2Aw++%2B+c%2Aw%5E2%29%29+
= +w%5E2+ [ +%28a+%2B+b+%2Aw++%2B+c+%2Aw%5E2%29 is cancelled ]
4.prove 1+(omega)^n+(omega)^2n=0 when n=2,4
Proof: Since +w%5E2+=+cos+%284%2Api%2F3%29+%2B+i+sin+%284%2Api%2F3%29+
= +cos+%28pi%2B+pi%2F3%29+%2B+i+sin+%28pi%2Bpi%2F3%29
= +-cos+%28pi%2F3%29+-+i+sin+%28pi%2F3%29
and +w+=+cos+%282%2Api%2F3%29+%2B+i+sin+%282%2Api%2F3%29+
= +w+=+cos+%28pi-+pi%2F3%29+%2B+i+sin+%28pi-pi%2F3%29+
= +w+=+-cos+%28pi%2F3%29+%2B+i+sin+%28pi%2F3%29+
When n =2, +1%2B+w%5E2+%2B+w%5E4+=+1+%2B+w%5E2+%2B+w+
[Note: +w%5E4+=+w]
= +1%2B+w+%2B+w%5E2+=+1+-cos+%28pi%2F3%29+-cos+%28pi%2F3%29+
= +1-1%2F2+-1%2F2++=+0+
{or use ew is the primitive root of +1%2Bx%2Bx%5E2+=+0 directly]

Similarly, when n =4, +1%2B+w%5E4+%2B+w%5E8+=+1+%2B+w+%2B+w%5E2+=+0
[Note: +w%5E8+=+w%5E2]

Kenny