Question

Find the product of all the roots of (-i) to the power 1/6

Answer 1

Step-by-step explanation:

If you understand complex numbers...

Let z = (-1)^(1/6)

z

=

(

−

1

)

1

6

=> z^6 = -1

⇒

z

6

=

−

1

Hence this is the equation we can try to solve to find the value of z

z

-1 = e^(ipi )

−

1

=

e

i

π

=> z^6 = e^(ipi)

⇒

z

6

=

e

i

π

We know e^(2kipi ) =1

e

2

k

i

π

=

1

AA k in ZZ

Multiply by this, is just the same as multiplying by 1:

=> z^6 = e^(ipi) * e^(2kipi)

=> z^6 = e^(ipi (1+2k) )

=> z = e^( (ipi)/6 ( 1+2k) ) , k = { 0,1,2,3,4,5 }

As all the other k yields repeated roots..

Hope its help..