`((x)/(3)+(1)/(x))^(5)` का मान ज्ञात कीजिए । इसका प्रयोग करके या अन्यथा `(sqrt(2)+1)^(6)+(sqrt(2)-1)^(6)` का मान ज्ञात कीजिए ।

Explanation

`(x+1)^(6)+(x-1)^(6)`
`=sum_(r=0)^(6).^(6)C_(r)x^(6-r)*(1)^(r)+sum_(r=0)^(6).^(6)C_(r)x^(6-r)(-1)^(r)`
`=(.^(6)C_(0)*x^(6)+^(6)C_(1)*x^(5)+^(6)C_(2)*x^(4)+^(6)C_(3)*x^(3)+^(6)C_(4)*x^(2)+^(6)C_(5)*x+^(6)C_(6))+(.^(6)C_(0)*x^(6)-.^(6)C_(1)*x^(5)+^(6)C_(2)*x^(4)-.^(6)C_(3)*x^(3)+^(6)C_(4)*x^(2)-.^(6)C_(5)*x+^(6)C_(6))`
`=2*[.^(6)C_(0)*x^(6)+^(6)C_(2)*x^(4)+^(6)C_(4)*x^(2)+^(6)C_(6)]`
`=2(x^(6)+15x^(4)+15x^(2)+1)`
`x=sqrt(2)` रखने पर
`(sqrt(2)+1)^(6)+(sqrt(2)-1)^(6)`
`=2[(sqrt(2))^(6)+15(sqrt(2))^(4)+15(sqrt(2))^(2)+1]`
`=2[8+60+30+1]=198`