Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents. (4a^2b^−1)^−2(4a^−2b)^−2
To simplify the expression (4a^2b^-1)^-2(4a^-2b)^-2, we can apply the property of exponents which states that when raising a power to a negative exponent, we can invert the base and change the exponent to positive.
Expanding the first portion, (4a^2b^-1)^-2 becomes (1/(4a^2b^-1))^2.
Expanding the second portion, (4a^-2b)^-2 becomes (1/(4a^-2b))^2.
Now, we can further simplify each of these expressions:
(1/(4a^2b^-1))^2 = (1^2)/(4^2)(a^2)^2(b^-1)^2
= 1/(16a^4b^-2)
= 1/(16a^4/b^2)
= b^2/(16a^4)
(1/(4a^-2b))^2 = (1^2)/(4^2)(a^-2)^2(b^1)^2
= 1/(16a^-4b^2)
= 1/(16/a^4b^2)
= a^4b^2/16
Finally, multiplying these two simplified expressions:
(b^2/(16a^4)) * (a^4b^2/16) = (b^2 * a^4b^2) / (16 * 16 * a^4)
= (a^4b^4) / (256a^4)
= b^4 / 256
Therefore, the simplified expression is b^4 / 256.