Simplifying Expressions with Negative Exponents: (9m^4n^10/3m^2n^5)^1
This article will explore the process of simplifying the expression (9m^4n^10/3m^2n^5)^1. We'll delve into the rules of exponents and how they apply to this specific case.
Understanding the Rules of Exponents
The simplification of the expression relies on several fundamental rules of exponents:
 Negative Exponent: A term raised to a negative exponent is equivalent to its reciprocal raised to the positive version of that exponent.
Example: x^2 = 1/x^2  Division of Exponents with the Same Base: When dividing terms with the same base, subtract the exponents. Example: x^m / x^n = x^(mn)
 Exponent of a Quotient: To raise a quotient to an exponent, raise both the numerator and denominator to that exponent. Example: (a/b)^n = a^n / b^n
Simplifying the Expression StepbyStep

Addressing the Negative Exponent: We begin by applying the negative exponent rule. (9m^4n^10/3m^2n^5)^1 = 1/ (9m^4n^10/3m^2n^5)

Simplifying the Inner Expression: Now, we simplify the expression within the denominator. 1/ (9m^4n^10/3m^2n^5) = 1 / (3m^2n^5)

Simplifying the Entire Expression: Finally, we combine the numerator and denominator and apply the rule for dividing exponents with the same base. 1 / (3m^2n^5) = (1/3)m^2n^5

Expressing with Positive Exponents: To express the result with positive exponents, we apply the negative exponent rule again. (1/3)m^2n^5 = (1/3) * (1/m^2) * (1/n^5) = m^2n^5/3
Final Result
Therefore, the simplified form of the expression (9m^4n^10/3m^2n^5)^1 is (1/3)m^2n^5 or m^2n^5/3. It's essential to understand and utilize the rules of exponents when simplifying expressions involving them.