SOLUTION: If {{{ (3/2 + isqrt(3)/2)^50 = 3^25(x + iy) }}} where x and y are real numbers then find x and y? Note: Here sqrt(3) means square root of 3.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: If {{{ (3/2 + isqrt(3)/2)^50 = 3^25(x + iy) }}} where x and y are real numbers then find x and y? Note: Here sqrt(3) means square root of 3.      Log On


   

Question 1184218: If +%283%2F2+%2B+isqrt%283%29%2F2%29%5E50+=+3%5E25%28x+%2B+iy%29+ where x and y are real numbers then find x and y?
Note: Here sqrt(3) means square root of 3.
Found 2 solutions by ikleyn, robertb:
Answer by ikleyn(52856) About Me  (Show Source):
You can put this solution on YOUR website!
.
If +%283%2F2+%2B+isqrt%283%29%2F2%29%5E50+=+3%5E25%28x+%2B+iy%29+ where x and y are real numbers then find x and y?
Note: Here sqrt(3) means square root of 3.
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The number  3%2F2+%2B+i%2A%28sqrt%283%29%2F2%29  has the modulus  r = sqrt%28%283%2F2%29%5E2+%2B+3%2F4%29 = sqrt%289%2F4+%2B+3%2F4%29 = sqrt%2812%2F4%29 = sqrt%283%29

                        and the argument  a = pi%2F6.


THEREFORE,  3%2F2+%2B+i%2A%28sqrt%283%29%2F2%29 = sqrt%283%29%2Acis%28pi%2F6%29.


Hence,  the left side number  %283%2F2+%2B+isqrt%283%29%2F2%29%5E50 is equal to  %28sqrt%283%29%29%5E50%2Acis%2850pi%2F6%29 = 3%5E25%2Acis%282pi%2F6%29 = 3%5E25%2Acis%28pi%2F3%29.


It implies that  x + iy = cis%28pi%2F3%29 = 1%2F2+%2B+i%2A%28sqrt%283%29%2F2%29.


So  x = 1%2F2,  y = sqrt%283%29%2F2.      ANSWER

Solved.



Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!


=
By comparing the last term in the chain of equality with 3%5E25%28x+%2B+iy%29+, it follows that x+=+1%2F2 and y+=+sqrt%283%29%2F2.