-81b^6c^3 : (3b^2c)^4

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To simplify this expression, we first need to expand the cube of the binomial (3b^2c)^4 using the power of a power rule:

(3b^2c)^4 = 3^4 (b^2)^4 c^4
= 81 b^8 c^4

Now, we substitute this back into the original expression:

-81b^6c^3 / (3b^2c)^4
= -81b^6c^3 / (81b^8c^4)

Now, we can simplify this expression by dividing each term in the numerator by the corresponding term in the denominator:

= -1/b^2

Therefore, the simplified expression is -1/b^2.