Respuesta :
The value of z^3 is option (D) [tex]z^{3}=64 (cos(\frac{3\pi }{2} ) + isin (\frac{3\pi }{2} ) )[/tex] is the correct answer.
What is a complex number?
A complex number is characterized by an ordered pair of numbers, the real and imaginary parts. It can be represented in multiple ways such as Standard form(rectangular form), Polar form, Exponential form.
For the given situation,
The complex number, z = 4 (cos (pi/ 2) + i sin (pi/2 ) )
[tex]z=4(cos(\frac{\pi }{2} ) + isin (\frac{\pi }{2} ) )[/tex]
⇒ [tex]z^{3}=4^{3} (cos(\frac{\pi }{2} ) + isin (\frac{\pi }{2} ) )^{3}[/tex]
By De Moivre's theorem,
[tex]{\displaystyle {\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx}[/tex]
Thus, [tex]z^{3}=64 (cos(\frac{3\pi }{2} ) + isin (\frac{3\pi }{2} ) )[/tex]
Hence we can conclude that the value of z^3 is option (D) [tex]z^{3}=64 (cos(\frac{3\pi }{2} ) + isin (\frac{3\pi }{2} ) )[/tex] is the correct answer.
Learn more about complex numbers here
https://brainly.com/question/24222471
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