Solution #14

Recall that, using the Binomial Theorem, the expansion of (x+y)ⁿ is the sum of all terms of the form nCk*x^n-ky^k.

We want a term in the expansion of  (a-1/ √ a  )⁷ in which the a term has an exponent of -1/2. Using the above general form, the k^th term will have an exponent of 7-k (from the a term) plus (-1/2)k (from the 1/ √ a  term). So we need 7 - k - k/2 = -1/2, or 15/2 = 3k/2, or k = 5. The coefficient is then (-1)⁵*7C5 = -21.