SOLUTION: I need help with: Write the three cube roots of i. Express the roots in rectangular form. Use exact values.
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Question 982703: I need help with: Write the three cube roots of i. Express the roots in rectangular form. Use exact values.
Found 2 solutions by ikleyn, solver91311:
Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website!
has 3 values:
1) = = cos(30°) + i*sin(30°) = ;
2) = cos(30°+120°) + i*sin(30° + 120°) = cos(150°) + i*sin(150°) = ;
3) = cos(30°+240°) + i*sin(30° + 240°) = cos(270°) + i*sin(270°) = = .
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
While ikleyn's solution using De Movire's Theorem has more general application, here is another method that is more specific to this particular problem that has the advantage of simple elegance:
Consider that ^3\ =\ (-i)(-i)^2\ =\ (-i)(-1)\ =\ i)
. Then we can say:
^3\ \Rightarrow\ z^3\ -\ (-i)^3\ =\ 0)
Next, recall the factorization of the difference of two cubes:
\right)\left(z^2\ +\ z(-i)\ +\ (-i)^2\right)\ =\ 0)
which simplifies to:
\right)\left(z^2\ -\ zi\ -\ 1\right)\ =\ 0)
So, if
\ =\ 0)
then
but if
then
}}{2}\ =\ \frac{i\ \pm\ \sqrt{3}}{2})
John
My calculator said it, I believe it, that settles it
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