Final Answer:
The expression ((9x)/(y))^(3) with positive exponents only simplifies to (729x^3)/(y^3).
Step-by-step explanation:
To simplify ((9x)/(y))^(3) with positive exponents only, we utilize the rule that states when raising a fraction to a power, you raise both the numerator and denominator to that power separately. In this case, raising (9x)/(y) to the power of 3 means cubing both the numerator and the denominator. Thus, we get:
(9x)^(3) / (y)^(3)
Expanding the numerator (9x)^(3) results in (9^3) * (x^3) = 729x^3. Similarly, the denominator (y)^(3) remains unchanged as y^3. Therefore, the simplified expression becomes (729x^3) / (y^3).
This process is based on the rule of exponents that states when a fraction is raised to a power, each part of the fraction (numerator and denominator) is separately raised to that power. Applying this rule in this case led to the simplification of the expression by raising 9x and y individually to the power of 3.
In summary, by cubing both the numerator and denominator separately, we obtained the expression (729x^3) / (y^3) as the simplified form of ((9x)/(y))^(3) with positive exponents only. This simplification adheres to the fundamental rules of exponents governing fractional exponents.