all the real cube roots of 2016

Question

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Answer 1

using DeMoivre's theorem, you can find THREE cube roots of any complex number.

 

In the case of 2016 + 0i one of the cube roots is exactly as Andy has found--the only Real cube root. 

 

The others are 2016^(1/3)(cos 210^o + i sin 210^o) and similar expression using angle 330^o

 

These can be found as exact values using knowledge of trigonometry and 30-60-90 right triangles as Reference Angle diagrams.

Answer 2

2016 = 2 x 1008 = 2 x 2 x 504 = 2 x 2 x 2 x 252 = 2 x 2 x 2 x 2 x 126 = 2 x 2 x 2 x 2 x 2 x 63

                                                                                                        = 2 x 2 x 2 x 2 x 2 x 7 x 9

                                                                                                        = 2 x 2 x 2 x 2 x 2 x 7 x 3 x 3 

 

so the cube root of 2016 is 2 * cube-root( 4 * 7 * 9)

                                         = 2 * cube-root( 252) = 12.632719195312758.....