`(sqrt(3)+sqrt(2))^(6)-(sqrt(3)-sqrt(2))^(6)` का मान ज्ञात कीजिए।

Explanation

द्विपद प्रमेय के विस्तार से,
`(sqrt(3)+sqrt(2))^(6)=.^(6)C_(0)(sqrt(3))^(6)+^(6)C_(1)(sqrt(3))^(5)sqrt(2)+^(6)C_(2)(sqrt(3))^(4)(sqrt(2))^(2)+^(6)C_(3)(sqrt(3))^(3)(sqrt(2))^(3)+^(6)C_(4)(sqrt(3))^(2)(sqrt(2))^(4)+^(6)C_(5)sqrt(3)(sqrt(2))^(5)+^(6)C_(6)(sqrt(2))^(6)`
और `(sqrt(3)-sqrt(2))^(6)=.^(6)C_(0)(sqrt(3))^(6)-^(6)C_(1)(sqrt(3))^(5)sqrt(2)+^(6)C_(2)(sqrt(3))^(4)(sqrt(2))^(2)-^(6)C_(3)(sqrt(3))^(3)(sqrt(2))^(3)+^(6)C_(4)(sqrt(3))^(2)(sqrt(2))^(4)-^(6)C_(5)sqrt(3)(sqrt(2))^(5)+^(6)C_(6)(sqrt(2))^(6)`
`therefore(sqrt(3)+sqrt(2))^(6)-(sqrt(3)-sqrt(2))^(6)=2[.^(6)C_(1)(sqrt(3))^(5)sqrt(2)+^(6)C_(3)(sqrt(3))^(3)(sqrt(2))^(3)+^(6)C_(5)(sqrt(3))(sqrt(2))^(5)]`
`=2[6xx9sqrt(3)xxsqrt(2)+20xx3sqrt(3)xx2sqrt2+6xxsqrt(3)xx4sqrt(2)]`
`=2[54sqrt(6)+120sqrt(6)+24sqrt(6)]`
`=2xx198sqrt(6)`
`=396sqrt(6)`
अतः `(sqrt(3)+sqrt(2))^(6)-(sqrt(3)-sqrt(2))^(6)=396sqrt(6)`