Question

Which expression represents a cube root of 1 + I?

Answer 1

Answer:

D (I think!!)

Step-by-step explanation:

I searched for a complex plane graph and put in each answer and this was the closest.

Answer 2

Answer:

$\sqrt[6]{2}$ [Cos ((3π/4)+ i Sin((3π/4))]

Step-by-step explanation:

Step 1, find modulus or radius:   r = $\sqrt{ 1^{2} + 1^{2} }$ =$\sqrt{2}$

Step 2 find theta:   Θ = tan ⁻¹ (1/1) = π/4

Step 3 write in polar form:   $\sqrt{2}$ [Cos (π/4) + i Sin(π/4)]

Step 4, use roots of complex number formula:  

$\sqrt[n]{r}$[Cos ((θ+2π)/(n)) + i Sin((θ+2π)/(n))]

Note:  n=3 (cube root) and r= $\sqrt{2}$ so $\sqrt[n]{r}$ equals $\sqrt[3]{\sqrt{} 2}$ = $\sqrt[6]{2}$

so, $\sqrt[6]{2}$ [ Cos ((π/4+2π)/3)) + i Sin((π/4+2π)/(3))]

     $\sqrt[6]{2}$ [Cos ((π/4+8π/4)/3)) + i Sin((π/4+8π/4)/(3))]

     $\sqrt[6]{2}$ [Cos ((9π/4)/3)+ i Sin((9π/4)/3))]

     $\sqrt[6]{2}$ [Cos ((9π/12)+ i Sin((9π/12))]

  = $\sqrt[6]{2}$ [Cos ((3π/4)+ i Sin((3π/4))]