Question

46. Find the value of a2+Va1aVa2 -1

Answer 1

The given expression is ( a 2 + a 2 −1 ) 4 + ( a 2 − a 2 −1 ) 4 .

Using the Binomial theorem to expand ( x+y ) 4 and ( x+y ) 4 ,

( x+y ) 4 = C 4 0 ( x ) 4−0 + C 4 1 ( x ) 4−1 y+ C 4 2 ( x ) 4−2 y 2 + C 4 3 ( x ) y 3 + C 4 4 y 4 = x 4 +4 x 3 y+6 x 2 y 2 +4x y 3 + y 4 (1)

( x−y ) 4 = C 4 0 ( x ) 4−0 − C 4 1 ( x ) 4−1 y+ C 4 2 ( x ) 4−2 y 2 − C 4 3 ( x ) y 3 + C 4 4 y 4 = x 4 −4 x 3 y+6 x 2 y 2 −4x y 3 + y 4 (2)

Therefore, the summation of the equation (1) and (2) is,

( x+y ) 4 + ( x−y ) 4 =2( x 4 +6 x 2 y 2 + y 4 )

Substitute the value of x= a 2 and y= 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 −6 a 4 + a 4 −2 a 2 +1 ] =2 a 8 +12 a 6 −10 a 4 −4 a 2 +2

Therefore, the value of ( a 2 + a 2 −1 ) 4 + ( a 2 − a 2 −1 ) 4 is 2 a 8 +12 a 6 −10 a 4 −4 a 2 +2 .