Find `(x+a)^(5)+(x-a)^(5)`. Hence, evaluate `(sqrt(6)+sqrt(7))^(5)+(sqrt(6)-sqrt(7))^(5)`

Explanation

We know that
`(x+a)^(n)=.^(n)C_(0)x^(n)+.^(n)C_(a)x^(n-1)a+.^(n)C_(2)x^(n-2)a^(2)+ . . .+.^(n)C_(r)x^(n-r)a^(r)+ . . . .+.^(n)C_(n)a^(n)` . . . (1)
`therefore(x+a)^(5)+(x-a)^(5)={.^(5)C_(0)x^(5)+.^(5)C_(1)x^(4)a+.^(5)C_(2)x^(3)a^(2)+.^(5)C_(3)x^(2)a^(3)+.^(5)C_(4)xa^(4)+.^(5)C_(5)a^(5)}`
`+{.^(5)C_(0)x^(5)-.^(5)C_(1)x^(4)a+.^(5)C_(2)x^(3)a^(2)-.^(5)C_(3)x^(2)a^(3)+.^(5)C_(4)xa^(4)-.^(5)C_(5)a^(5)}`
`=2{.^(5)C_(0)x^(5)+.^(5)C_(2)x^(2)a^(2)+.^(5)C_(4)xa^(4)}`
Thus `(a+x)^(5)+(x-a)^(5)=2{.^(5)C_(0)x^(5)+.^(5)C_(2)x^(3)a^(2)+.^(5)C_(4)xa^(4)}` . . . (2)
Now putting the value of x and a as `sqrt(6) and sqrt(7)`respectively, we get
`(sqrt(6)+sqrt(7))^(5)+(sqrt(6)+sqrt(7))^(5)`
`=2{.^(5)C_(0)(sqrt(6))^(5)+.^(5)C_(2)(sqrt(6))^(3)(sqrt(7))^(2)+.^(5)C_(4)(sqrt(6))(sqrt(7))^(4)}`
`=2{36sqrt(6)+10xx6sqrt(6)xx7+5xx49sqrt(6)}`
`=2{36sqrt(6)+420sqrt(6)+245sqrt(6)}`
`=2{701sqrt(6)}`
`=1406sqrt(6)`
Hence, `(sqrt(6)+sqrt(7))^(5)+(sqrt(6)-sqrt(7))^(5)=1402sqrt(6)`