Simplifying (9m^4n^10/3m^2n^5)^1
This problem involves simplifying an expression with negative exponents and fractions. Let's break it down stepbystep:
Understanding the Properties of Exponents
 Negative Exponents: A term raised to a negative exponent is equivalent to its reciprocal raised to the positive version of that exponent. For example, x^2 = 1/x^2.
 Fractions with Exponents: When a fraction is raised to an exponent, both the numerator and denominator are raised to that exponent. For example, (a/b)^2 = a^2/b^2.
 Product of Powers: When multiplying powers with the same base, add the exponents. For example, x^m * x^n = x^(m+n).
 Quotient of Powers: When dividing powers with the same base, subtract the exponents. For example, x^m / x^n = x^(mn).
Simplifying the Expression

Apply the negative exponent rule: (9m^4n^10/3m^2n^5)^1 = 1 / (9m^4n^10/3m^2n^5)

Simplify the fraction inside the parentheses: 1 / (9m^4n^10/3m^2n^5) = 3m^2n^5 / 9m^4n^10

Simplify the coefficients and apply the quotient of powers rule: 3m^2n^5 / 9m^4n^10 = 1/3 * m^(24) * n^(510)

Calculate the exponents: 1/3 * m^(24) * n^(510) = 1/3 * m^2 * n^5

Apply the negative exponent rule again: 1/3 * m^2 * n^5 = 1/3 * 1/m^2 * 1/n^5

Combine the terms: 1/3 * 1/m^2 * 1/n^5 = 1/(3m^2n^5)
Conclusion
Therefore, the simplified form of (9m^4n^10/3m^2n^5)^1 is 1/(3m^2n^5).