Question

The coefficient of x^(-5)=?, in the binomial expansion of [((x+1)/((x^(2/3)-(x^(1/3))+1))-((x-1)/(x-(x^(1/2))]^10 , where 'x' not equal to 0,1 is:

Answer 1

1

Explanation:

Since `a^3+1 = (a+1)(s^2-a+1)`, we have
`x+1 = (x^{1/3}+1)(x^{2/3}-x^{1/3}+1)`, so that
`{x+1}/{x^{2/3}-x^{1/3}+1}=x^{1/3}+1`
Again
`{x-1}/{x-x^{1/2}}={(x^{1/2}-1)(x^{1/2}+1)}/{x^{1/2}(x^{1/2}-1)}= 1+x^{-1/2}`

Thus the given expression simplifies to

`(x^{1/3}-x^{-1/2})^10`

The only way that we can get `x^{-5}` here is from the term
`.^10C_10(x^{1/3})^0 (-x^{-1/2})^10` - and the coefficient of this term is `(-1)^10 = 1`