Need help with a question about powers of complex numbers

In summary: Hello, @CoolKid223.Welcome to our class. Unfortunately, we do not know how you are supposed to solve this problem. Are you supposed to know about the polar coordinate representation of complex numbers?Or maybe you are using logz , to express ## z^a = e^{a log z} ##, restricting to branches?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 = -
  • #1
CoolKid223
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0
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.
 
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  • #2
CoolKid223 said:
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...
 
  • #3
Welcome, @CoolKid223!
##\sqrt[3]{-8}=-2##
Hmm...Does it help?
Greetings
 
  • #5
CoolKid223 said:
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 .

:welcome:

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.
 
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  • #6
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.
 
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  • #7
CoolKid223 said:
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?
 
  • #8
Or maybe you are using logz , to express ## z^a = e^{a log z} ##, restricting to branches?
 
  • #9
WWGD said:
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.
 
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Related to Need help with a question about powers of complex numbers

1. What are complex numbers?

Complex numbers are numbers that consist of a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part with i being the imaginary unit.

2. How do you calculate powers of complex numbers?

To calculate powers of complex numbers, you can use De Moivre's theorem which states that (a + bi)^n = r^n(cos(nθ) + i sin(nθ)), where r is the modulus of the complex number and θ is the argument of the complex number.

3. What is the difference between a real number and a complex number?

A real number is a number that can be represented on a number line and does not have an imaginary component. A complex number, on the other hand, has both a real and an imaginary component.

4. How do you simplify complex numbers?

To simplify a complex number, you need to combine like terms and use the rules of arithmetic for complex numbers. This may involve adding, subtracting, multiplying, or dividing the real and imaginary parts of the complex number.

5. What are the applications of complex numbers?

Complex numbers have many applications in mathematics, physics, engineering, and other fields. They are used to study electrical circuits, analyze vibrations and waves, and solve differential equations, among other things.

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