Math Help - Show me the steps to solving complex numbers involving e?

1. 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=

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

Find the product of and conjugate Z?

How do I solve to get the answer 27?

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

Hello, INeedOfHelp!

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

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

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

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

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

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

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

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

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

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