Answer:
Correct answer: a) -8
Step-by-step explanation:
Complex Numbers
A complex number z in polar form is expressed as
[tex]z=r(\cos\theta+\mathbf{i}\sin\theta)[/tex]
The nth power of a complex number in polar form is:
[tex]z^n=r^n(\cos(n\theta)+\mathbf{i}\sin(n\theta))[/tex]
If the complex number is given in rectangular form
[tex]z=x+\mathbf{i}y[/tex]
then the values of r and θ are given by:
[tex]r=\sqrt{x^2+y^2}[/tex]
[tex]\displaystyle \theta=\arctan\left(\frac{y}{x}\right)[/tex]
The given complex number is:
[tex]z=1-\mathbf{i}\sqrt{3}[/tex]
The value of r is:
[tex]r=\sqrt{1^2+(-\sqrt{3})^2}[/tex]
[tex]r=\sqrt{1+3}=2[/tex]
r=2
Calculating the angle:
[tex]\displaystyle \theta=\arctan\left(\frac{-\sqrt{3}}{1}\right)[/tex]
[tex]\displaystyle \theta=\arctan(-\sqrt{3})[/tex]
Since the number is in the quadrant IV, the angle is:
[tex]\theta=-60^\circ[/tex]
Thus, z is expressed in polar form as:
[tex]z=2(\cos(-60^\circ)+\mathbf{i}\sin(-60^\circ))[/tex]
Calculating:
[tex]z^3=2^3(\cos(-180^\circ)+\mathbf{i}\sin(-180^\circ))[/tex]
Given cos 180°=-1 and sin(-180°)=0
[tex]z^3=8(-1+0)[/tex]
[tex]z^3=-8[/tex]
Correct answer: a) -8
