Final answer:
The fourth term in the expansion of (a - b)^8 using the Binomial theorem is 56a5b^3.
Step-by-step explanation:
The question asks us to find the fourth term in the expansion of (a - b)^8 using the Binomial theorem. According to the binomial theorem, any expression of the form (a + b)^n expands as follows:
an + nan-1b + n(n-1)an-262 / 2! + n(n-1)(n-2)an-363 / 3! + ...
Since we are looking for the fourth term, we need to use the third term in the formula, which corresponds to n(n-1)(n-2)an-363 / 3!, because the index starts at zero.
For (a - b)^8, n = 8 and our formula for the fourth term becomes:
T4 = 8(8-1)(8-2)a8-353 / 3!
= 8 * 7 * 6 * a5(-b)3 / 3!
= 56a5(-b)^3
Simplifying the factorial, we find that the fourth term of the expansion is:
T4 = 56a5b^3.