Final answer:
To simplify (125x⁵y⁴)/(5x⁸y⁶) using only positive exponents, the expression becomes 25/x³y² which represents dividing 125 by 5 to get 25, and subtracting the exponents of like variables, taking reciprocals as needed to ensure all exponents are positive.
Step-by-step explanation:
To fully simplify the expression (125x⁵y⁴)/(5x⁸y⁶) using only positive exponents, we'll apply the laws of exponents to simplify both the numerical coefficients and the variables separately. For the numerical part, since 125 is equal to 5³, this simplifies with the 5 in the denominator, leaving us with 5² or 25 in the numerator. For the variables, we subtract exponents when dividing like bases: x⁵/x⁸ gives us x⁻³ (since 5 - 8 = -3), and y⁴/y⁶ gives us y⁻² (since 4 - 6 = -2). To express these with positive exponents, we take reciprocal, resulting in x³ in the denominator and y² in the denominator:
= (5³ × x⁵ × y⁴)/(5 × x⁸ × y⁶)
= (25 × x⁵−⁸ × y⁴−⁶)
= (25/x³y²)
The final answer is 25/x³y².