Simplify. Assume that no denominator is equal to zero.
(4a^-1)^-2/(2a^4)^2
(4a^-1)^-2 / (2a^4)^2 = 4^-2a^2 / 4a^8 = 4^-3a^-6 = 1 / 4^3a^6 = 1 / 64a^6.
To simplify the expression (4a^-1)^-2/(2a^4)^2, we need to apply the properties of exponents and simplify each part separately.
First, let's simplify the numerator, (4a^-1)^-2:
1. Apply the power of a power rule, which states that (a^m)^n = a^(m * n):
(4a^-1)^-2 = 4^(-2) * (a^-1)^-2
2. Simplify the exponent by applying the power of a product rule, which states that (ab)^n = a^n * b^n:
(4a^-1)^-2 = 4^(-2) * a^(-1 * -2)
3. Simplify the exponents and evaluate 4^(-2):
(4a^-1)^-2 = 1/4^2 * a^2
(4a^-1)^-2 = 1/16 * a^2
Now let's simplify the denominator, (2a^4)^2:
1. Apply the power of a product rule:
(2a^4)^2 = 2^2 * (a^4)^2
2. Simplify the exponent:
(2a^4)^2 = 4 * a^(4 * 2)
(2a^4)^2 = 4 * a^8
Now that we simplified the numerator and the denominator, let's write the simplified expression:
(4a^-1)^-2 / (2a^4)^2 = (1/16 * a^2) / (4 * a^8)
To divide by 4, we can multiply by its reciprocal:
(4a^-1)^-2 / (2a^4)^2 = (1/16 * a^2) * (1/(4 * a^8))
Simplifying further:
(4a^-1)^-2 / (2a^4)^2 = 1/(16 * 4 * a^2 * a^8)
(4a^-1)^-2 / (2a^4)^2 = 1/(64 * a^10)
Therefore, the simplified expression is 1/(64 * a^10).