Final answer:
The original expression simplifies to 81/(16x⁸y⁸) after applying laws of exponents and arithmetic operations. All exponents are positive, as per the requirements of the question.
Step-by-step explanation:
To simplify the expression ((3x⁵) / (2x⁷y²))^4 using only positive exponents, we first start by simplifying within the parentheses using the law of exponents, which states that when you divide like bases, the exponents should be subtracted. The exponent in the parentheses becomes x²y². We also simplify the number to be 3/2. Therefore, the expression simplifies to be (3/2)^4 * (1/x²y²)^4.
Next, we apply the exponent of 4 to each term inside parentheses which yields 81/16 * 1/x⁸y⁸.
Finally, the expression simplifies to be 81/(16x⁸y⁸), expressing everything in the positive exponents as the question asks.
Learn more about Simplifying Exponential Expressions