Question

Evaluate : (2+√3)⁷ + (2-√3)⁷

Answer 1

To find: Value of (2+√3)⁷ + (2-√3)⁷

Formula used: (i)

ⁿC_r = \(\frac{n!}{(n-r)!(r)!}\)

(ii) (a+b)ⁿ = ⁿC₀aⁿ + ⁿC₁a^n-1b + ⁿC₂a^n-2b² + … +ⁿC_n-1ab^n-1 + ⁿC_nbⁿ 

⇒ 2[(1)a⁷ + (21)a⁵b² + (35)a³b⁴ + (7)ab⁶] 

⇒ 2[a⁷ + 21a⁵b² + 35a³b⁴ + 7ab⁶] = (a+b)⁷ + (a-b)⁷ 

Putting the value of a = 2 and b = √3 in the above equation

(2+√3)⁷ + (2-√3)⁷

=  2[{2⁷} + {21(2)⁵(√3)²} + {35(2)³(√3)⁴} + {7(2)(√3)⁶}]

= 2[128 + 21(32)(3)+ 35(8)(9) + 7(2)(27)] 

= 2[128 + 2016 + 2520 + 378] 

= 10084