Final answer:
To simplify the expression (3a^2b^-6)^-4, raise each term inside the parentheses to the -4th power, including the coefficient and variables. The simplified expression is 1/(81a^8b^24).
Step-by-step explanation:
To simplify the expression (3a^2b^-6)^-4, we need to raise each term inside the parentheses to the -4th power, including both the coefficient and the variables. The coefficient 3 becomes 3^-4 = 1/81. The term a^2 becomes a^(2 * -4) = a^-8 = 1/a^8. The term b^-6 becomes b^(-6 * -4) = b^24.
Putting it all together, the simplified expression is (1/81)(1/a^8)(b^24), which can be rewritten as 1/(81a^8b^24).