Question

x=a+b, y= a alpha + b beta, z= a beta + b alpha, where alpha and beta are complex cube roots of unity, then prove that
xyz= a^3 + b^3

Answer 1

the answer is a+b×a alpha + b beta × a beta + b alpha = a^alpha 3 + b^beta 3 = ab^alpha ^beta 6

Answer 2

xyz  = a³ + b³  if x   = a  + b , y = aα  + bβ  , z = aβ + bα  where α & β  are complex cube roots of unity

Step-by-step explanation:

x   = a  + b

y = aα  + bβ

z = aβ + bα

α & β  are complex cube roots of unity

=> α * β = 1   &   α + β = -1

xyz  = (a + b) (aα  + bβ)(aβ + bα)

= (a + b) ( a²αβ  + abα² + abβ² + b²αβ)

= (a + b) (αβ(a² + b²) + ab(α² + β²))

=(a + b)( a² + b²  - ab)

= a³ + ab² - a²b  + ba² + b³ - ab²

= a³ + b³

QED

Proved

xyz  = a³ + b³

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