simplify (a^-2b^2/a^2b^-1)^-3

To simplify this expression, we can first apply the negative exponent rules.

(a^-2b^2/a^2b^-1)^-3 = (b^2/a^2b^-1)^-3 / (a^-2b^2)^-3

Next, we can simplify the part of the expression in the numerator: (b^2/a^2b^-1)^-3

Using the exponent rule for division, this becomes: (b^2)^-3 / (a^2b^-1)^-3

Simplifying further:

(b^2)^-3 is equal to b^(-6) (since raising a power to a negative exponent is the same as taking the reciprocal of the base raised to the positive exponent)

(a^2b^-1)^-3 is equal to (a^2)^-3 * (b^-1)^-3

(a^2)^-3 is equal to a^(-6)

(b^-1)^-3 is equal to b^(-(-3)) = b^3 (since a negative exponent raised to a negative exponent becomes positive)

Putting it all together:

(b^2/a^2b^-1)^-3 / (a^-2b^2)^-3 = (b^-6) / (a^-6 * b^3)

Next, we can apply the exponent rule for division by subtracting the exponents:

(b^-6) / (a^-6 * b^3) = b^(-6-3) / a^-6

Simplifying the exponents:

b^-9 / a^-6

Finally, we can apply the exponent rule for taking the reciprocal of a power:

b^-9 is equal to 1 / b^9

a^-6 is equal to 1 / a^6

(b^-9) / (a^-6) = (1 / b^9) / (1 / a^6)

Using the rule for dividing by a fraction is the same as multiplying by its reciprocal:

(1 / b^9) / (1 / a^6) = (1 / b^9) * (a^6 / 1)

Simplifying the expression:

(1 * a^6) / (b^9 * 1) = a^6 / b^9

Therefore, (a^-2b^2/a^2b^-1)^-3 simplifies to a^6 / b^9.