Question

Let \(\omega\) be the cube root of unity. Then the expression \[\Large \prod_{k=0}^{11}(x+(-1)^{k-2} \iota \space \omega^{2k-5})\] where x=20 is

Answer 1

options are \[1). (20^6+1)^2\]\[2). (20^6-1)^2\]\[3). (20^3+1)^4\]\[4). (20^3-1)^4\]

Answer 2

@ganeshie8

Answer 3

As a start, the given expression is same as

\[ \prod_{k=0}^{11}(x+(-1)^{k-2} i \space \omega^{2k-5})\\~\\
= \prod_{k=0}^{11}(x+(i^2)^{k-2} i \space \omega^{2k-5})\\~\\
= \prod_{k=0}^{11}(x-i^{2k-5} \space \omega^{2k-5})\\~\\
= \prod_{k=0}^{11}(x- (i\omega)^{2k-5})\\~\\
\]

Answer 4

Im thinking maybe plug in some value of k and see if there is a pattern

Answer 5

we know that the cube roots are\[1,\omega,\omega ^{2}\]

Answer 6

if k is 0, then\[(i \omega)^{-5}=-i*\omega ^{-2}\]

Answer 7

I gt it it is (20^6+1)^2