Question

11. Find (a by - (a - by. Hence, evaluate 32)- (3 - V2)

Answer 1

The given expression ( a+b ) 4 − ( a−b ) 4 and we have to evaluate ( 3 + 2 ) 4 − ( 3 − 2 ) 4 .

The formula for binomial expansion is ,

( a+b ) n = C n 0 a n + C n 1 a n−1 b+ C n 2 a n−2 b 2 +..........+ C n n−1 a. b n−1 + C n n b n

The expression ( a+b ) 4 − ( a−b ) 4 , can be expanded as

According to the question n=4

` ( a+b ) 4 = C 4 0 a 4 b 0 + C 4 1 a 4−1 b 1 + C 4 2 a 4−2 b 2 + C 4 3 a 4−3 b 3 + C 4 4 a 4−4 b 4 = C 4 0 a 4 + C 4 1 a 3 b 1 + C 4 2 a 2 b 2 + C 4 3 a 1 b 3 + C 4 4 b 4 (1)

( a−b ) 4 = C 4 0 a 4 b 0 − C 4 1 a 4−1 b 1 + C 4 2 a 4−2 b 2 − C 4 3 a 4−3 b 3 + C 4 4 a 4−4 b 4 = C 4 0 a 4 − C 4 1 a 3 b 1 + C 4 2 a 2 b 2 − C 4 3 a 1 b 3 + C 4 4 b 4 (2)

Subtract equation (2) from equation (1),

( a+b ) 4 − ( a−b ) 4 =[ [ C 4 0 a 4 + C 4 1 a 3 b 1 + C 4 2 a 2 b 2 + C 4 3 a 1 b 3 + C 4 4 b 4 ] −[ C 4 0 a 4 − C 4 1 a 3 b 1 + C 4 2 a 2 b 2 − C 4 3 a 1 b 3 + C 4 4 b 4 ] ] =2( C 4 1 a 3 b+ C 4 3 a b 3 ) =2( 4 a 3 b+4a b 3 ) =8ab( a 2 + b 2 ) (3)

Substitute the value of a= 3 and b= 2 in equation (3), we get

( 3 + 2 ) 4 − ( 3 − 2 ) 4 =8( 3 )( 2 ){ ( 3 ) 2 + ( 2 ) 2 } =8( 6 ){ 3+2 } =40 6

Thus the evaluation of the expression ( 3 + 2 ) 4 − ( 3 − 2 ) 4 is 40 6 .