Let [z = ( 3 2 + i 2 )^5 + ( 3 2 - i 2 )^5.] If R(z) and 1(z) respectively denote the real…

Question

Let $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5}.$$ If R(z) and 1(z) respectively denote the real and imaginary parts of z, then :

Accepted Answer

$$z = {\left( {{{\sqrt 3  + i} \over 2}} \right)^5} + {\left( {{{\sqrt 3  - i} \over 2}} \right)^5}$$

$$z = {\left( {{e^{i\pi /6}}} \right)^5} + {\left( {{e^{ - i\pi /6}}} \right)^5}$$

$$ = {e^{i5\pi /6}} + {e^{ - i5\pi /6}}$$

$$ = \cos {{5\pi } \over 6} + i{{\sin 5\pi } \over 6} + \cos \left( {{{ - 5\pi } \over 6}} \right) + i\sin \left( {{{ - 5\pi } \over 6}} \right)$$

$$ = 2\cos {{5\pi } \over 6} < 0$$

$${\rm I}(z) = 0$$ and  $${\mathop{\rm Re}\nolimits} (z) < 0$$

Solutions · 1