find the fifth roots of -32i

Answer 1

Notice that -32i = 2%5E5%2Acis%28270%5Eo%29.

Therefore, due to DeMoivre formulas, the 5-th degree roots of -32i are

(1) 2cis(54°); ( notice that 54° = 270%5Eo%2F5 );

(2) 2cis%2854%5Eo+%2B+360%5Eo%2F5%29 = = 2cis(54° + 72°) = 2cis(126°); ( Notice that 72° = )

(3) 2cis%2854%5Eo+%2B+2%2A%28360%5Eo%2F5%29%29 = 2cis(54° + 2*72°) = 2cis(198°);

(4) 2cis%2854%5Eo+%2B+3%2A%28360%5Eo%2F5%29%29 = 2cis(54° + 3*72°) = 2cis(270°);

(5) 2cis%2854%5Eo+%2B+4%2A%28360%5Eo%2F5%29%29 = 2cis(54° + 4*72°) = 2cis(342°).