Final answer:
To simplify the expression ((3a^5 c)^2/(2a^3 b)^3)^2, we use the power to a power rule and the rule for dividing terms with the same base. The final expression is simplified to (9a^2c^4)/(8b^6), which corresponds to answer choice B.
Step-by-step explanation:
The question asks us to simplify the expression ((3a^5 c)^2/(2a^3 b)^3)^2 using the property of exponents.
First, let's apply the power to a power rule, which states that when we raise a power to another power, we multiply the exponents. So the expression inside the parentheses will be simplified as follows:
- (3a^5 c)^2 becomes 3^2 * a^(5*2) * c^2 which is 9a^10c^2
- (2a^3 b)^3 becomes 2^3 * a^(3*3) * b^3 which is 8a^9b^3
Now our expression looks like this: (9a^10c^2)/(8a^9b^3).
Next, we simplify by dividing the terms with the same base, reducing the exponents where applicable. By dividing a^10 by a^9, we cancel out 9 a's, leaving us with just a. So the simplified expression is now (9a^1c^2)/(8b^3).
Finally, we need to apply the power outside the parentheses, raising everything to the power of 2. We get:
- 9^2 becomes 81
- a^1 squared becomes a^2
- c^2 squared becomes c^4
- 8 squared becomes 64
- b^3 squared becomes b^6
However, since 81/64 is not fully simplified, we know that both numbers can be divided by their greatest common divisor, which is 1. So the final simplified form of the expression is (9a^2c^4)/(8b^6).