What is the solution for (1-i)^n = -512 - 512i?

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In summary, the conversation discusses a problem involving solving (1-i)^n = -512 - 512i for n, with the solution being n = 19. The conversation also touches on the issue of posting homework problems in the general math section, with one participant suggesting that these types of problems should be posted in a designated thread in order to avoid cluttering the forums.
  • #1
mathwizarddud
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Solve [tex](1-i)^n = -512 - 512i[/tex] for n.
:wink:
 
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  • #2
mathwizarddud said:
Solve [tex](1-i)^n = -512 - 512i[/tex] for n.
:wink:

as in you need help with it? Or a problem for us to do?

Anyhow...it looks relatively simple.
 
  • #3
I'm thinking n = 19
 
  • #4
Even I can do that one.
 
  • #5
The fact that [itex](\sqrt{2})^10= 512[/itex] makes that pretty trivial.
 
  • #6
mathwizarddud said:
Solve [tex](1-i)^n = -512 - 512i[/tex] for n.
:wink:
What is the point of this thread?
 
  • #7
mathwizarddud, many of us seem to be slightly irritated you constantly post these problems here in the *Homework Help* section. You may think these problems are interesting, and I'm sure they interest some other people as well, but please, post these in the general math section, all under a single thread, named something obvious like "Simple Math Problems" or alike.
 
  • #8
Gib Z, whether or not the question is a real homework question is irrelevant. If it's a standard textbook question, it belongs in the HW forums, and the OP is required to show original effort. Otherwise the Math forums just become a backdoor to avoiding our Guidelines for homework.
 

What does the equation (1-i)^n = -512 - 512i mean?

The equation (1-i)^n = -512 - 512i is a complex number equation, where n represents the exponent and i represents the imaginary unit. This equation is asking for the value of n that will result in the given complex number.

What is the solution to the equation (1-i)^n = -512 - 512i?

The solution to this equation is n = 10. This can be found by using the complex number formula (a+bi)^n = r^n(cos(nθ) + i sin(nθ)), where a and b are the real and imaginary parts of the complex number, r is the modulus, and θ is the argument. In this case, a = 1, b = -1, r = √2, and θ = -π/4. Plugging these values into the formula gives (1-i)^n = (√2)^n(cos(-nπ/4) + i sin(-nπ/4)). Setting this equal to -512 - 512i and solving for n gives n = 10.

What does the solution n = 10 represent in the equation (1-i)^n = -512 - 512i?

The solution n = 10 represents the power or exponent needed to raise (1-i) to in order to get the complex number -512 - 512i. In other words, (1-i)^10 = -512 - 512i.

How can I check if n = 10 is the correct solution to the equation (1-i)^n = -512 - 512i?

You can check this by plugging n = 10 into the equation and verifying that the result is -512 - 512i. For example, (1-i)^10 = (√2)^10(cos(-10π/4) + i sin(-10π/4)) = (√2)^10(cos(-5π/2) + i sin(-5π/2)) = (√2)^10(0 + i(-1)) = -(√2)^10 = -512 - 512i. Therefore, n = 10 is the correct solution.

Are there any other solutions to the equation (1-i)^n = -512 - 512i?

Yes, there are infinitely many other solutions to this equation. This is because for any integer k, (1-i)^(10+4k) also equals -512 - 512i. This is because (1-i)^10 = (√2)^10(cos(-10π/4) + i sin(-10π/4)) = -(√2)^10 = -512 - 512i, and multiplying this by (1-i)^4 will not change the result. Therefore, n = 10+4k, where k is any integer, is also a solution to the equation.

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