To simplify the expression 24x^5y^4/13ab^2 : 4xy^2/13a^2 b : 3x^2(y-2)/a^2b, we will first express each term as a fraction and then divide the fractions.

First term: 24x^5y^4/13ab^2 can be written as a fraction with a numerator of 24x^5y^4 and a denominator of 13ab^2.

Second term: 4xy^2/13a^2 b can be written as a fraction with a numerator of 4xy^2 and a denominator of 13a^2 b.

Third term: 3x^2(y-2)/a^2b can be written as a fraction with a numerator of 3x^2(y-2) and a denominator of a^2b.

Now, let's divide the fractions:

(24x^5y^4/13ab^2) / (4xy^2/13a^2 b) / (3x^2(y-2)/a^2b)

To divide fractions, we multiply by the reciprocal of the second fraction and then the third fraction:

(24x^5y^4/13ab^2) (13a^2 b/4xy^2) (a^2b/3x^2(y-2))

Now, simplify:

24x^5y^4 13a^2 b a^2b / 13ab^2 4xy^2 3x^2(y-2)
312a^2 x^5 y^4 b^2 / 52ab^2 x y^2 9x^2(y-2)
6a x^4 y^2 / x(y-2)
6ax^3/ (y-2)