2 Attachment(s)

Show me the steps to solving complex numbers involving e?

My teacher didn't go into too much detail on these equations and they were on a test, so I'd like to know how to solve these equations for my final.

Z= Attachment 26072

Can someone show me the steps how to get this complex number into algebraic form answer Z=4i ?

Find the product of Attachment 26073 and conjugate Z?

How do I solve to get the answer 27?

Re: Show me the steps to solving complex numbers involving e?

Hello, INeedOfHelp!

Quote:

$\displaystyle z \,=\,4e^{\frac{\pi}{2}i}$

Can someone show me the steps how to get this complex number into algebraic form: $\displaystyle z 4i$

$\displaystyle z \:=\:4e^{\frac{\pi}{2}i} \;=\;4\left(\cos\tfrac{\pi}{2} + i\sin\tfrac{\pi}{2}\right) \;=\; 4(0 + i\!\cdot\!1) \;=\;4i$

Quote:

Find the product of $\displaystyle z \,=\,3\;\!\sqrt{3}e^{\pi i} $ and its conjugate $\displaystyle \overline{z}$

How do I solve to get the answer $\displaystyle 27$?

We have: .$\displaystyle \begin{Bmatrix}z &=& 3\sqrt{3}\;\!e^{\pi i} \\ \\[-3mm] \overline{z} &=& 3\sqrt{3}\;\!e^{-\pi i} \end{Bmatrix}$

Therefore: .$\displaystyle z\cdot\overline{z} \;=\;\left(3\sqrt{3}\;\!e^{\pi i}\right)\left(3\sqrt{3}\;\!e^{-\pi i}\right) \;=\; \left(3\sqrt{3}\cdot3\sqrt{3}\right)\left(e^{\pi i}\cdot e^{-\pi i}\right) $

n . . . . . . . . . . . $\displaystyle =\;(27)(e^0) \;=\;(27)(1) \;=\;27$

Re: Show me the steps to solving complex numbers involving e?

Oh well that's easy enough, thank you very much.