Factorise a^7 - ab^6.

To factorize the expression a^7 - ab^6, follow these steps:

Step 1: Look for common factors in both terms. The common factor between a^7 and ab^6 is 'a'. So, we can factor out 'a' from both terms.

a^7 - ab^6 = a(a^6 - b^6)

Step 2: Factorize the difference of squares in the remaining expression. a^6 - b^6 is a difference of squares, which can be written as:

a^6 - b^6 = (a^3 - b^3)(a^3 + b^3)

Step 3: Recognize that a^3 - b^3 is a difference of cubes. We can apply the formula for the difference of cubes, which states that:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

So, the factorization of the entire expression is:

a^7 - ab^6 = a(a - b)(a^2 + ab + b^2)(a^3 + b^3)

This is the fully factorized form of the given expression.