We know that- Hence = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 Thus, (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 Replacing b with -b (a + (-b)) 4 = a 4 + 4a 3 (-b) + 6a 2 (-b) 2 + 4a(-b) 3 + (-b) 4 (a-b) 4 = a 4 - 4a 3 b + 6a 2 b 2 - 4ab 3 + b 4 Now, (a + b) 4 + (a - b) 4 = (a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 ) + (a 4 - 4a 3 b + 6a 2 b 2 - 4ab 3 + b 4 ) = 2(a 4 + 6a 2 b 2 + b 4 ) Putting a = a 2 & b = √(a 2 -1) (a 2 + √(a 2 -1)) 4 + (a 2 - √(a 2 -1)) 4 = 2[(a 2 ) 4 + 6(a 2 ) 2 (√(a 2 -1)) 2 + (√(a 2 -1)) 4 ] = 2[ a 8 + 6(a 4 )(a 2 -1) + (a 2 -1) 2 ] = 2[ a 8 + 6 a 6 - 6a 4 + a 4 - 2 a 2 + 1] = 2[ a 8 + 6 a 6 - 5a 4 - 2 a 2 + 1]