Consider the complex number z=(i^201+i^8)/(i^3(1+i)^2).
(a) Show that z can be expressed in the Cartesian form 1/2+(1/2)i.
(b) Find the modulus of 4z − 2z*. (z* meaning z-bar/complex conjugate of z)
(c) Write 2z in polar form.
(d) Write 8z^3 in polar exponential form.
(c) = 1+i, r = √2, θ=π/4
(d) = -2-2i
The Attempt at a Solution
(a) I have already done.
(b) I ended up with 1+3i (is this right by the way?), with √1^2+3^2 = √10. So, does the modulus equal √10? However, I entered it into wolfram alpha and got 1-3i instead. So, which one is right?
(c) 2z=√2(cos(π/4)+isin(π/4)). -- Is that right? (Also with wolfram alpha, I got 2z=1-i (cartesian form) so, θ=-π/4 -- is this right instead?)
(d) I am confused with. How would I write out 8z^3? Would I first have to use it with (1/2)+(1/2)i and then write that given equation in polar exponential form? I tried that, with getting -2-2i and therefore, √8e^iπ/4. Is that correct?