Question

Use DeMoivre's Theorem to find the twelfth (12th) power of the complex number, and write result in standard form?

2[cos(LaTeX: \frac{\pi}{2} π 2 ) + i sin(LaTeX: \frac{\pi}{2} π 2 )]

Answer 1

`( 2[cos( \frac{\pi}{2}) + i sin( \frac{\pi}{2} ) ] )^{12} = 4096`

Explanation:

I think the questioner is asking for

`( 2[cos( \frac{\pi}{2}) + i sin( \frac{\pi}{2} ) ] )^{12}`

using DeMoivre.

`( 2[cos( \frac{\pi}{2}) + i sin( \frac{\pi}{2} ) ] )^{12}`

`= 2^{12} (cos(pi/2) + i sin(pi/2))^12`

`= 2^{12} (cos(6 pi) + i sin (6pi) )`

`= 2^12 (1 + 0 i)`

`= 4096`

Check:

We don't really need DeMoivre for this one:

`cos(pi/2) + i \sin(pi/2) = 0 + 1i = i`

`i^12 = (i^4)^3= 1^3=1`

so we're left with `2^{12}.`