Question

Find the fifth roots of 32(cos 280° + i sin 280°)

Answer 1

The fifth root is 2[cos(56°) + i sin(56°)]

Explanation:

* To solve this problem we must revise De Moiver's rule

- In the complex number with polar form

∵ z = r(cosФ + i sinФ)

∴ z^n = r^n(cos(nФ) + i sin(nФ))

* In the problem

- The fifth root means z^(1/5)

- We can put 32 as a form a^n

∵ 32 = 2 × 2 × 2 × 2 × 2 = 2^5

∴ z = 2^5[cos(280°) + i sin(280°)]

* Lets find z^(1/5)

`*z^{(1)/(5)}=[2^(5)]^{(1)/(5) } (cos((1)/(5))(280)+isin((1)/(5))(280)`

`*(2^(5))^{(1)/(5)}=2^{5.(1)/(5)}=2`

∴ z^(1/5) = 2[cos(56) + i sin(56)]

* The fifth root of 32[cos(280°) + i sin(280°)] is 2[cos(56°) + i sin(56°)]