Assuming x to be so small that x^2 and higher powers of x can be neglected, the coefficient of x in (1-x)^ 1 / 3 +(1-5 x)^2 (16-x)^ 1 / 4 is equal to

Step-by-step solution

Given expression aligned & (1-x)^ 1 / 3 +(1-5 x)^2 (16-x)^ 1 / 4 = & 1 2 [(1-x)^ 1 / 3 (1- x 16 )^ -1 / 4 +(1-5 x)^2 (1- x 16 )^ -1 / 4 ] = & 1 2 [ (1- 1 3 x ) (1+ x 64 )+(1-10 x) (1+ x 64 ) ] aligned (On ignoring the higher degree terms) = 1 2 (1- x 3 + x 64 +1-10 x+ x 64 ) (On ignoring the higher degree terms) = 1 2 (2-x ( 1 3 +10- 1 32 ) ) Coefficient of x=- 32+960-3 192 =- 989 192