Omega

Algebra Level 3

What is the value of ( 1 − ω + ω 2 ) 4 + ( 1 + ω − ω 2 ) 4 ? \large(1 - \omega + \omega^2)^4 + (1 + \omega - \omega^2)^4 ?

ω \omega denotes the primitive cube roots of unity.

-32 -16 256 4

Naren Bhandari by Naren Bhandari , 21, India

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1 solution

Md Zuhair
Feb 24, 2017

An easy one, We can use ω 2 + ω + 1 = 0 {\omega}^2+\omega + 1 = 0

( 1 − ω + ω 2 ) 4 + ( 1 + ω − ω 2 ) 4 = ( − 2 ω ) 4 ( − 2 ω 2 ) 4 \large(1 - \omega + \omega^2)^4 + (1 + \omega - \omega^2)^4 = (-2\omega)^4 (-2\omega^2)^4 = 16 ω 4 + 16 ω 8 = − 16 16\omega^4 + 16\omega^8 = \boxed{-16}

4 Helpful 0 Interesting 0 Brilliant 1 Confused

@Naren Bhandari FYI An imaginary number is one of the form b i bi for some real number b b . You are referring to complex numbers, which are of the form a + b i a + bi .

Calvin Lin Staff - 4 years, 3 months ago

Sir, I'm extremely Sorry for the mis-information. I mean to mention cube roots of unity.

Naren Bhandari - 4 years, 3 months ago

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