Question

For `z^5 = 1 + i`, solve the complex number?

Answer 1

`z_0 = root[10]{2}(cos frac{pi}{20} + i sin frac{pi}{20})`
`z_1 = root[10]{2}(cos frac{9pi}{20} + i sin frac{9pi}{20})`
`z_2 = root[10]{2}(cos frac{17pi}{20} + i sin frac{17pi}{20})`
`z_3 = root[10]{2}(cos frac{25pi}{20} + i sin frac{25pi}{20})`
`z_4 = root[10]{2}(cos frac{33pi}{20} + i sin frac{33pi}{20})`

Explanation:

`z^5 = sqrt 2 (sqrt2 /2 + i sqrt 2/2)`

`z^5 = sqrt 2 (cos frac{pi}{4} + i sin frac{pi}{4})`

#z = root[10]{2}(cos frac{pi/4 + 2kpi}{5} + i sin frac{pi/4 + 2kpi}{5}); k = 0, 1, 2, 3, 4#

`z = root[10]{2}(cos frac{pi+8kpi}{20} + i sin frac{pi+8kpi}{20}); k = 0, 1, 2, 3, 4`