Simplifying ((1/(4ab^2c))^2)/((3/(2a^2bc^2))^4)

To simplify this expression, we need to use the rules of exponents and fractions.

Rule 1: To raise a fraction to a power, we can raise both the numerator and denominator to the power.

Rule 2: To raise a power to another power, we can multiply the exponents.

Rule 3: To divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction.

Step 1: Simplify the first fraction by squaring both the numerator and denominator:

(1/(4ab^2c))^2 = 1^2/(4^2a^2b^4c^2) = 1/(16a^2b^4c^2)

Step 2: Simplify the second fraction by raising both the numerator and denominator to the fourth power:

(3/(2a^2bc^2))^4 = 3^4/(2^4a^8b^4c^8) = 81/(16a^8b^4c^8)

Step 3: Rewrite the expression as a division of the first fraction by the second fraction, by multiplying the first fraction by the reciprocal of the second fraction:

((1/(4ab^2c))^2)/((3/(2a^2bc^2))^4) = (1/(16a^2b^4c^2))/(81/(16a^8b^4c^8))

= (1/(16a^2b^4c^2)) * (16a^8b^4c^8/81)

Step 4: Simplify the expression by cancelling out factors that appear in both the numerator and denominator:

= (1/81) * (a^6/c^6)

Step 5: Final Simplification:

(1/(4ab^2c))^2/((3/(2a^2bc^2))^4) = a^6/c^6 * 1/81

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