Solving Homework Questions: Modulus, Arg, Roots

[King_Silver]

Homework Statement

First question = attached below
Second Question = attached below

Homework Equations

modulus, argument, roots.

The Attempt at a Solution

For the 1st question I've gotten the modulus, the principal argument and the abs. of z^1/4 which was something like 1.5335 I believe. The only thing I am not understanding with regards to this question is the final part, where it is asking for the smallest argument, 2nd smallest, 2nd largest and largest in that ascending order.
I've tried several things but none of them have worked to give me a proper answer. I'm probably missing something really stupid like "Just divide it by x to find smallest, add x to find largest, take away x to find second smallest" or some standard format like that.

2nd question I got these as my answers
1: e^(3x)
2: -3y

3: -(i*arg(-2+4i))/3 + ln(abs(-2+4i))/3

yet they don't seem to be the correct answers, I think I must be leaving out a step in the 2nd one.

 

[SammyS]

Homework Statement

First question = attached below
Second Question = attached below

Homework Equations

modulus, argument, roots.

The Attempt at a Solution

For the 1st question I've gotten the modulus, the principal argument and the abs. of z^1/4 which was something like 1.5335 I believe. The only thing I am not understanding with regards to this question is the final part, where it is asking for the smallest argument, 2nd smallest, 2nd largest and largest in that ascending order.
I've tried several things but none of them have worked to give me a proper answer. I'm probably missing something really stupid like "Just divide it by x to find smallest, add x to find largest, take away x to find second smallest" or some standard format like that.

2nd question I got these as my answers
1: e^(3x)
2: -3y

3: -(i*arg(-2+4i))/3 + ln(abs(-2+4i))/3

yet they don't seem to be the correct answers, I think I must be leaving out a step in the 2nd one.

I suggest that these should be posted two threads, one thread for each problem. They are not all that similar, and each is multi-part.

It would be helpful to post the images as full-size like this:

 

[SteamKing]

Homework Statement

First question = attached below
Second Question = attached below

Homework Equations

modulus, argument, roots.

The Attempt at a Solution

For the 1st question I've gotten the modulus, the principal argument and the abs. of z^1/4 which was something like 1.5335 I believe. The only thing I am not understanding with regards to this question is the final part, where it is asking for the smallest argument, 2nd smallest, 2nd largest and largest in that ascending order.
I've tried several things but none of them have worked to give me a proper answer. I'm probably missing something really stupid like "Just divide it by x to find smallest, add x to find largest, take away x to find second smallest" or some standard format like that.

When you are asked to calculate something like z^1/n, you should be aware that there are n separate and distinct complex numbers, call them w₁, w₂, w₃, ..., w_n, which result. These complex numbers are related to what are known as the roots of unity, which is described here:

https://en.wikipedia.org/wiki/Root_of_unity

If you express the original complex number z in its polar form, such that z = r e ^{i θ}, then the various values of z^1/n can be easily computed.

 

[Ray Vickson]

Homework Statement

First question = attached below
Second Question = attached below

Homework Equations

modulus, argument, roots.

The Attempt at a Solution

For the 1st question I've gotten the modulus, the principal argument and the abs. of z^1/4 which was something like 1.5335 I believe. The only thing I am not understanding with regards to this question is the final part, where it is asking for the smallest argument, 2nd smallest, 2nd largest and largest in that ascending order.
I've tried several things but none of them have worked to give me a proper answer. I'm probably missing something really stupid like "Just divide it by x to find smallest, add x to find largest, take away x to find second smallest" or some standard format like that.

2nd question I got these as my answers
1: e^(3x)
2: -3y

3: -(i*arg(-2+4i))/3 + ln(abs(-2+4i))/3

yet they don't seem to be the correct answers, I think I must be leaving out a step in the 2nd one.

You should avoid posting images only. Type out the problem if it is not long and/or complicated. Type out you solution also, to the extent possible. As Vela states in his post "Guidelines for students and helpers": "Ask yourself "If I can't be bothered to spend my time typing it, why should they be bothered to spend their time reading it, much less responding to it?" Use images for supporting figures."