Question

A)The cube roots of
1 + i
B)Give the exact roots. (Express θ in radians.)

Answer 1

http://www.wolframalpha.com/input/?i=%281+%2B+i%29^%281%2F3%29

Answer 2

Really, @cebroski ?

You will need DeMoivre, right? Show it and demonstrate it.

Answer 3

@tkhunny
By DeMoivre,
\[Z ^{N} = R ^{N}( \cos(N \theta) + i \sin(N \theta))\]

Answer 4

where Z is a complex number, R is the modulus, N is a rational number, sin and cos are trigonometric functions and i = sqrt(-1)

Answer 5

Let Z = 1 + i, N = 1, so that we may solve for R and theta.
\[Z^1 = 1 + i = R^1(\cos \theta + i \sin \theta)\]
For what values of theta and R make the above equation true?
\[\theta = \frac{ -7\pi }{ 4 }, \frac{ \pi }{ 4 }, \frac{ 9\pi }{ 4 }\]
\[R = \sqrt {2}\]

Answer 6

Now, let us find the cube root of 1 + i. Let Z = 1 + i, N = 1/3,
\[R = \sqrt {2}\]
\[\theta = \frac{ -7\pi }{ 4 }, \frac{ \pi }{ 4 }, \frac{ 9\pi }{ 4 }\]