(9m^4n^10/-3m^2n^5)^-1 Simplified

3 min read Jun 16, 2024

Simplifying (9m^4n^10/-3m^2n^5)^-1

This problem involves simplifying an expression with negative exponents and fractions. Let's break it down step-by-step:

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^(m-n).

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^(2-4) * n^(5-10)
Calculate the exponents: -1/3 * m^(2-4) * n^(5-10) = -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)

Therefore, the simplified form of (9m^4n^10/-3m^2n^5)^-1 is -1/(3m^2n^5).