Question

Find the exact values of all three cube roots of 1 and then add them. Do the same for the fourth, fifth, sixth, and eighth roots of 1. What do you think is the sum of the n th roots of 1 , for any n ?

Answer 1

The cube roots, fourth roots, fifth roots, sixth roots, and eighth roots of 1 and their sums are explained.

Step-by-step explanation:

To find the cube roots of 1, we need to find numbers a such that a³ = 1. The cube roots of 1 are 1, -0.5 + 0.866i, and -0.5 - 0.866i, where i is the imaginary unit. Adding these cube roots gives us 1 + (-0.5 + 0.866i) + (-0.5 - 0.866i) = 0.

For the fourth, fifth, sixth, and eighth roots of 1, we can follow a similar process. The fourth roots of 1 are 1, i, -1, and -i. The fifth roots of 1 are 1, cos(72°) + isin(72°), cos(144°) + isin(144°), cos(216°) + isin(216°), and cos(288°) + isin(288°), where cos and sin are trigonometric functions. The sixth roots of 1 are 1, cos(60°) + isin(60°), cos(120°) + isin(120°), cos(180°) + isin(180°), cos(240°) + isin(240°), and cos(300°) + isin(300°).

The eighth roots of 1 are 1, cos(45°) + isin(45°), cos(90°) + isin(90°), cos(135°) + isin(135°), cos(180°) + isin(180°), cos(225°) + isin(225°), cos(270°) + isin(270°), and cos(315°) + isin(315°).

For the sum of the nth roots of 1, the sum will always be 0 because for every root a, there is a corresponding root -a, which cancels out when added together.