Let = - 1 + i 3 2 and = - 1 - i 3 2 , i = - 1 . If ( 7 - 7 + 9 ) 20 + ( 9 + 7 - 7 ) 20 + ( - 7 + 9 + 7 ) 20 + ( 14 + 7 + 7 ) 20 = m 10 , then m is [2026]

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Solution

Q.
Let $α = \frac{- 1 + i \sqrt{3}}{2} and β = \frac{- 1 - i \sqrt{3}}{2}, i = \sqrt{- 1} .$

If ${(7 - 7 α + 9 β)}^{20} + {(9 + 7 α - 7 β)}^{20} + {(- 7 + 9 α + 7 β)}^{20} + {(14 + 7 α + 7 β)}^{20} = m^{10},$ then $m$ is __________ . [2026]

Ans.
(49)

$(9 + 7 ω - 7 ω^{2}) + ω^{20} {(9 + 7 ω - 7 ω^{2})}^{20} + ω^{40} {(9 + 7 ω - 7 ω^{2})}^{20} + (14 + 7 {(ω + ω^{2}))}^{20}$

${(9 + 7 ω - 7 ω^{2})}^{20} (1 + ω + ω^{2}) + {(14 - 7)}^{20}$

$= 7^{20}$

$= {(49)}^{10}$

$Hence, M = 49$

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