Question

# If $$\omega$$ is an imaginary cube root of unity then find: $$(1-\omega+\omega^2)(1-\omega^2+\omega^4)(1-\omega^4+\omega^8)$$ ..... to 2n factors $$=$$

A
2n+1
B
22n
C
22n+1
D
2n

Solution

## The correct option is B $$2^{2n}$$$$(1-w+{ w }^{ 2 })(1-{ w }^{ 2 }+{ w }^{ 4 })(1-{ w }^{ 4 }+{ w }^{ 8 })...2n\quad terms\\ =(1-w+{ w }^{ 2 })(1-{ w }^{ 2 }+w)(1-w+{ w }^{ 2 })...\\ =(0-2w)(0-2{ w }^{ 2 })(0-2w)(0-2{ w }^{ 2 })...2n\quad terms\\ ={ (-2w) }^{ n }{ (-2{ w }^{ 2 }) }^{ n }={ (2w.2{ w }^{ 2 }) }^{ n }={ 4 }^{ n }={ 2 }^{ 2n }\\$$Hence, (b) is correct.Maths

Suggest Corrections

0

Similar questions
View More

People also searched for
View More