Write the complex number z=(1-sqrt3i)^14 in polar form: z=r(costheta+i sin theta) where r = and theta =

Question

Please help ME

Answer 1

You conflate two different formats "polar form" and evolution of a complex by De Moivre's Theorem.

 

z = 1 - i√3

r = 2

θ = 5π/3

 

z¹⁴ = 2¹⁴(cos 70π/3 + i sin 70π/3)

Answer 2

The technique to use when raising a complex number to a power is to convert the number to polar form (r θ)

 

In this case the 1, and -sqrt3 i form a special triangle based in the quadrant IV.    Find the magnitude r using the pythagorean theorem.     r = 2.   Find the angle using trigonometry and place in correct quadrant.   θ = 5π/3.

 

So the problem becomes to  (2 5π/3)^14.     We find this value by raising the magnitude to the power 14 and multiplying the angle by 14.     (2^14  14*5π/3)  =  (16384  70π/3)   = (16384  4π/3)