सरल कीजिए `(x+sqrt(x-1))^(6)+(x-sqrt(x-1))^(6)`

Explanation

माना कि `sqrt(x-1)=a" ":.(x+sqrt(x-1))^(6)=(x+a)^(6)`
`=x^(6)+""^(6)C_(1)x^(5)a+""^(6)C_(2)x^(4)a^(2)+""^(6)C_(3)x^(3).a^(3)+""^(6)C_(4)x^(2)a^(4)+""^(6)C_(5)xa^(5)+""^(6)C_(6)a^(6)" "...(ii)`
`:." "(x+sqrt(x-1))^(6)+(x-sqrt(x-1))^(6)`
`=[x^(6)+""^(6)C_(1)x^(5)a+""^(6)C_(2)x^(4)a^(2)+""^(6)C_(3)x^(3)a^(3)+""^(6)C_(4)x^(2)a^(4)+""^(6)C_(5)xa^(5)+""^(6)C_(6)a^(6)]+[x^(6)-""^(6)C_(1)x^(5)a+""^(6)C_(2)x^(4)a^(2)-""^(6)C_(3)x^(3)a^(3)+""^(6)C_(4)x^(2)a^(4)-""^(6)C_(5)xa^(5)+""^(6)C_(6)a^(6)]`
`=2[x^(6)+""^(6)C_(2)x^(4)a^(2)+""^(6)C_(4)x^(2)a^(4)+""^(6)C_(6)a^(6)]`
`=2[x^(6)+15x^(4)a^(2)+15x^(2)a^(4)+a^(6)]`
`=2[x^(6)+15x^(4)(sqrt(x-1))^(2)+15x^(2)(sqrt(x-1))^(4)+(sqrt(x-1))^(6)]" "[a" का मान रखने पर"]`
`=2[x^(6)+15x^(4)(x-1)+15x^(2)(x-1)^(2)+(x-1)^(3)]`
`=2[x^(6)+15x^(5)-15x^(4)+15x^(4)-30x^(3)+15x^(2)+x^(3)-3x^(2)-3x-1]`
`=2[x^(6)+15x^(5)-29x^(3)+12x^(2)+3x-1]`