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- Homework Statement
- NIL

- Homework Equations
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Hello all!

Thanks for helping me out so far :) Really appreciate it.

I don't seem to understand some of the questions presented to me, so if anyone has an idea on how to start the questions, please do render your assistance :)

1)

Suppose

(a+bi)(c+di)(e+fi)=4+8i(a+bi)(c+di)(e+fi)=4+8i

Find the value of

(a2+b2)(c2+d2)(e2+f2)(a2+b2)(c2+d2)(e2+f2)

Not sure what I'm suppose to do here, expanding is probably out of the question, does squaring (a+bi)(c+di)(e+fi)(a+bi)(c+di)(e+fi) helps to find out (a2+b2)(c2+d2)(e2+f2)(a2+b2)(c2+d2)(e2+f2)?

2)

Let

S=(cos(π/5)+isin(π/5))n,nϵNS=(cos(π/5)+isin(π/5))n,nϵN

What I understand here is that I'm supposed to find the amount of distinct roots in this equation? How do I even start?

5)

Let ∣z1∣=∣z2∣=7∣z1∣=∣z2∣=7

If ∣z1+z2∣=2∣z1+z2∣=2,solve ∣1/z1+1/z2∣

How do I even proceed from here?

3)

Let z be complex number that allows:

z+7¯¯¯z=∣¯¯¯z+4∣∣z+7z¯=∣z¯+4∣

Find z.

My working:

a+bi+7(a−bi)=∣(a+4)+bi∣a+bi+7(a−bi)=∣(a+4)+bi∣

8a−6bi=√(a+4)2+b28a−6bi=√(a+4)2+b2

64a2−96abi−36b2=a2+8a+16+b264a2−96abi−36b2=a2+8a+16+b2

Not sure where to proceed from here.
4)

Take ##3+7i## is a solution of ##3x^2+Ax+B=0##

Since ##3+7i## is a solution, I can only gather :

##(z−(3+7i))(...)=3x2+Ax+B##

Not sure on how to go from here.

EDIT: I got A =18 and B=174, is this correct?

I recognized that since there's a 3, this means the other root must be a conjugate, hence

##(z-(3+7i))(z-(3-7i))##

##(z-3)^2-(7i)^2 =0##

##z^2+6z+58=0##

##3z^2+18z+174=0##

6)

Suppose ##z=2e^{ikπ}##and

##z^{n}=2^5 e^{iπ/8}##

Find k such that z has smallest positive argument

I don't understand this question :/ For z to have smallest positive principal argument, what does it entail/mean?

EDIT: Tried again. Got the following:

##z^{n}=2^n e^{inkπ} = 2^5 e^{iπ/8}##

## nk = 1/8##

##5k =1/8##

##k = 1/40##?

7)

Let

##\sum_{k=0}^9 x^k = 0##

Find smallest positive argument. Same thing as previous question, but I guess I can expand to

z+z2+z3+...+z9=0z+z2+z3+...+z9=0

##z=re^iθ##

##rei^θ+re^2iθ+re^3iθ+...##

What do I do to proceed on?

Cheers

Thanks for helping me out so far :) Really appreciate it.

I don't seem to understand some of the questions presented to me, so if anyone has an idea on how to start the questions, please do render your assistance :)

Suppose

(a+bi)(c+di)(e+fi)=4+8i(a+bi)(c+di)(e+fi)=4+8i

Find the value of

(a2+b2)(c2+d2)(e2+f2)(a2+b2)(c2+d2)(e2+f2)

Not sure what I'm suppose to do here, expanding is probably out of the question, does squaring (a+bi)(c+di)(e+fi)(a+bi)(c+di)(e+fi) helps to find out (a2+b2)(c2+d2)(e2+f2)(a2+b2)(c2+d2)(e2+f2)?

2)

Let

S=(cos(π/5)+isin(π/5))n,nϵNS=(cos(π/5)+isin(π/5))n,nϵN

What I understand here is that I'm supposed to find the amount of distinct roots in this equation? How do I even start?

5)

Let ∣z1∣=∣z2∣=7∣z1∣=∣z2∣=7

If ∣z1+z2∣=2∣z1+z2∣=2,solve ∣1/z1+1/z2∣

How do I even proceed from here?

3)

Let z be complex number that allows:

z+7¯¯¯z=∣¯¯¯z+4∣∣z+7z¯=∣z¯+4∣

Find z.

My working:

a+bi+7(a−bi)=∣(a+4)+bi∣a+bi+7(a−bi)=∣(a+4)+bi∣

8a−6bi=√(a+4)2+b28a−6bi=√(a+4)2+b2

64a2−96abi−36b2=a2+8a+16+b264a2−96abi−36b2=a2+8a+16+b2

Not sure where to proceed from here.

Take ##3+7i## is a solution of ##3x^2+Ax+B=0##

Since ##3+7i## is a solution, I can only gather :

##(z−(3+7i))(...)=3x2+Ax+B##

Not sure on how to go from here.

EDIT: I got A =18 and B=174, is this correct?

I recognized that since there's a 3, this means the other root must be a conjugate, hence

##(z-(3+7i))(z-(3-7i))##

##(z-3)^2-(7i)^2 =0##

##z^2+6z+58=0##

##3z^2+18z+174=0##

6)

Suppose ##z=2e^{ikπ}##and

##z^{n}=2^5 e^{iπ/8}##

Find k such that z has smallest positive argument

I don't understand this question :/ For z to have smallest positive principal argument, what does it entail/mean?

EDIT: Tried again. Got the following:

##z^{n}=2^n e^{inkπ} = 2^5 e^{iπ/8}##

## nk = 1/8##

##5k =1/8##

##k = 1/40##?

7)

Let

##\sum_{k=0}^9 x^k = 0##

Find smallest positive argument. Same thing as previous question, but I guess I can expand to

z+z2+z3+...+z9=0z+z2+z3+...+z9=0

##z=re^iθ##

##rei^θ+re^2iθ+re^3iθ+...##

What do I do to proceed on?

Cheers

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