Question

Write the complex number in Cartesian form: `e^((ln4-ipi)/4)` How do I do that?

Answer 1

`e^((ln 4 - ipi)/4) = 1-i`

Explanation:

Euler's formula tells us:

`e^(i theta) = cos theta + i sin theta`

So we find:

`e^((ln 4 - ipi)/4) = e^((ln 4)/4) * e^(-(ipi)/4)`

`color(white)(e^((ln 4 - ipi)/4)) = 4^(1/4)(cos(-pi/4) + i sin(-pi/4))`

`color(white)(e^((ln 4 - ipi)/4)) = sqrt(2)(sqrt(2)/2 - sqrt(2)/2 i)`

`color(white)(e^((ln 4 - ipi)/4)) = 1-i`