1. Plot and find the absolute value of z=5-3i
2. Write z=2sqrt( 2+i ) in trigonometic form
3.Use DeMoivre's Theorem to simplify (1+sqrt(3i))^6
Thank you for the help! :)
For problem 2, the answer is 1 + sqrt(2)*i.
For problem 3, we can use DeMoivre's theorem to simplify (1+sqrt(3i))^6. DeMoivre's theorem states that for any complex number z = r(cosθ + isinθ), where r is the modulus and θ is the argument, we have
(z)^n = r^n (cos nθ + isin nθ)
In this case, z = 1+sqrt(3i), so r = 1 and θ = arctan(sqrt(3)) = pi/3. Plugging these values into DeMoivre's theorem, we get
(1+sqrt(3i))^6 = 1^6 (cos 6θ + isin 6θ) = 1 (cos (6 * pi/3) + isin (6 * pi/3)) = 1 (-1 + 0i) = -1
Therefore, (1+sqrt(3i))^6 = -1.
