Simplify:

b^4(1/3b^2)(12b^-8)
(12x^2Y^4)(1/2x^5y)
(rs)^3(25)^-2(4r)^4

To simplify each expression, we'll use the properties of exponents.

1) Simplifying b^4(1/3b^2)(12b^-8):
a) The first step is to multiply the coefficients together, which gives us (1/3)(12) = 4.
b) Next, we multiply the bases with the same exponent. We add the exponents when multiplying with the same base. So, b^4 * b^2 = b^(4+2) = b^6.
c) Finally, we divide the bases with different exponents by subtracting the exponents. b^6 / b^8 = b^(6-8) = b^-2.

Therefore, the simplified expression is 4b^-2.

2) Simplifying (12x^2Y^4)(1/2x^5y):
a) Similar to the previous step, we multiply the coefficients: (12)(1/2) = 6.
b) Then, we multiply the bases with the same exponent. Here, the base x has different exponents, but we can subtract the exponents: x^(2+5) = x^7. The same applies to Y: Y^4 * Y^1 = Y^(4+1) = Y^5.
c) Finally, we multiply the remaining "y" in the parentheses: y * y = y^1 * y^1 = y^(1+1) = y^2.

Therefore, the simplified expression is 6x^7Y^5y^2.

3) Simplifying (rs)^3(25)^-2(4r)^4:
a) First, we simplify the bases within each set of parentheses. (rs)^3 = r^3s^3, (4r)^4 = 4^4r^4.
b) Next, we address the exponent of 25^-2. When the exponent is negative, we can rewrite the expression as the reciprocal: (1/25^2).
c) Now, we multiply the coefficients together: 1 * 4^4 = 256.
d) For the bases, we multiply the variables with the same base. r^3 * r^4 = r^(3+4) = r^7. The same goes for "s": s^3.
e) We multiply the remaining 25^-2 = 1/25^2.

Therefore, the simplified expression is 256r^7s^3/(25^2).