Question

If $α,β,γ$ are the cube roots of $-2$ ,then the value of $xα+yβ+zγ/xβ+yγ+zα$ is ($x,y,z$ are variables)

• A. $$$e^{ iπ/3}$
• B. $e^{2πi/3}$
• C. $1$
• D. $-1$
• E. $e^{ 4iπ/3}$

Answer 1

Answer: (E)

$e^{ 4iπ/3}$

Answer 2

$\dfrac{ xα + yβ + zγ }{xβ + yγ + zα }$

$=\dfrac{x(−2)^{1/3} + y (−2^{1/3}ω ) + z ( −2^{1/3}ω 2 )} {x (−2^{1/3}ω) + y (−2^{1/3}ω^2 ) + z(−2)^{1/3} }$

$= \dfrac{(−2)^{1/3}(x + yω + zω^2 )ω}{ −2^{1/3}(xω+ yω^2 +z) ω }$
$= 1 ω = \dfrac{ω ^3}{ω} = ω^2$
$= e^{4πi/3}$
So, the correct option is (E) : $e^{4πi/3}$