Help Me Solve $\text{Im}(z^3)$ in Rectangular Form

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Discussion Overview

The discussion revolves around finding the imaginary part of the expression $\text{Im}(z^3)$ for a complex number $z = x + jy$, specifically in rectangular form. Participants are exploring the algebraic manipulation required to isolate the imaginary component.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about how to proceed with finding $\text{Im}(z^3)$ and presents their current work.
  • Another participant reiterates the same question and asks for clarification on which terms contain the imaginary unit "j."
  • A third participant confirms that the imaginary part can be expressed as $(3x^2y - y^3)$, suggesting that the earlier expression is already simplified.
  • Subsequent posts challenge the correctness of the previous claims, with one participant asserting that the terms containing "j" are $3x^2yj - j^3$ and questioning if this is the answer.
  • Another participant corrects this assertion, reiterating that the imaginary part is indeed $3x^2y - y^3$ based on the definition of the imaginary component.

Areas of Agreement / Disagreement

There is disagreement regarding the identification of the imaginary part, with some participants asserting different expressions for $\text{Im}(z^3)$. The discussion remains unresolved as participants challenge each other's interpretations.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in isolating the imaginary part, and there are differing interpretations of the terms involved.

Drain Brain
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I don't how to proceed further

Let $z=x+jy$ find in rectangular form:

1. $\text{Im}(z^3)$

this is where I can get to,

$\text{Im}(x^3+3x^2jy+3xj^2y^2+j^3y^3)=\text{Im}(x^3+3x^2jy-3xy^2-jy^3)$

please help me.
 
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Drain Brain said:
I don't how to proceed further

Let $z=x+jy$ find in rectangular form:

1. $\text{Im}(z^3)$

this is where I can get to,

$\text{Im}(x^3+3x^2jy+3xj^2y^2+j^3y^3)=\text{Im}(x^3+3x^2jy-3xy^2-jy^3)$

please help me.

So what are the terms which have a "j" part?
 
Drain Brain said:
I don't how to proceed further

Let $z=x+jy$ find in rectangular form:

1. $\text{Im}(z^3)$

this is where I can get to,

$\text{Im}(x^3+3x^2jy+3xj^2y^2+j^3y^3)=\text{Im}(x^3+3x^2jy-3xy^2-jy^3)$

please help me.

Hi Drain Brain! :)

You're already there:
$$\text{Im}(x^3+3x^2jy-3xy^2-jy^3) = \text{Im}((x^3-3xy^2)+(3x^2y-y^3)j)
= (3x^2y-y^3)$$
 
they are $3x^2yj-j^3$ is this the answer?
 
Drain Brain said:
they are $3x^2yj-j^3$ is this the answer?

no .as I like serena mentioned it is
$3x^2y- y^3$ because imaginary part of $x+jy$ is $y $
 

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