Need help with a question about powers of complex numbers

[CoolKid223]

Homework Statement

Let (z-3)^3=-8. Solve for z, finding all real and complex solutions.

Relevant Equations

(z-3)^3=-8

(z-3)³=-8, solve for z.

I'm new to complex numbers, so I'm stuck on this basic problem: how do you find all real and non-real solutions in the equality, (z-3)^3=-8? Thanks a bunch.

 

[berkeman]

Homework Statement:: Let (z-3)^3=-8. Solve for z, finding all real and complex solutions.
Relevant Equations:: (z-3)^3=-8

(z-3)³=-8, solve for z.

I'm new to complex numbers, so I'm stuck on this basic problem: how do you find all real and non-real solutions in the equality, (z-3)^3=-8? Thanks a bunch.

Please do your best to start working the problem. Please show us what z is...

 

[mcastillo356]

Welcome, @CoolKid223!
$\sqrt[3]{-8}=-2$
Hmm...Does it help?
Greetings

 

[DaveE]

https://tutorial.math.lamar.edu/Extras/ComplexPrimer/ComplexNumbers.aspx

 

[SammyS]

Homework Statement:: Let (z-3)^3=-8. Solve for z, finding all real and complex solutions.
Relevant Equations:: (z-3)^3=-8

(z-3)³=-8, solve for z.

I'm new to complex numbers, so I'm stuck on this basic problem: how do you find all real and non-real solutions in the equality, (z-3)^3=-8? Thanks a bunch.

Hello @CoolKid223 .



You could try $(x+iy)^3=-8$ where $x$ and $y$ real real numbers. Then equate real parts and equate imaginary parts. Of course, $i^2 = -1$ .

Also, it seems this thread should be in the Pre-Calculus Forum.
I'll ask for it to be moved.

 

[DaveE]

My personal bias with complex numbers: always use the polar form whenever possible. Of course, sometimes that's not the best choice, but usually, for me, it is.

At least consider that there are two choices. Learn Euler's formula $e^{ix} = cos(x) + i⋅sin(x)$ and always keep it in mind. If the problem is addition and subtraction, use the rectangular form. If it's multiplication, division, powers, and roots, use the polar form.

 

[FactChecker]

Homework Statement:: Let (z-3)^3=-8. Solve for z, finding all real and complex solutions.
Relevant Equations:: (z-3)^3=-8

(z-3)³=-8, solve for z.

I'm new to complex numbers, so I'm stuck on this basic problem: how do you find all real and non-real solutions in the equality, (z-3)^3=-8? Thanks a bunch.

Unless we see some of your work, we do not know how your class is expected to solve problems. Are you supposed to know about the polar coordinate representation of complex numbers?

 

[WWGD]

Or maybe you are using logz , to express $z^a = e^{a log z}$, restricting to branches?

 

[Mark44]

Or maybe you are using logz , to express $z^a = e^{a log z}$, restricting to branches?

I doubt this is the case, since the OP said he is new to complex numbers.

The simplest approach, IMO, is to let w = z - 3, and then solve the equation $w^3 = -8$ for w. When those solutions are found, replace w by z - 3 in each of the three solutions for w.

In any case, we should let the OP wrestle with this problem a bit before offering any more suggestions.