Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 16m^-5n^4/12m^2n^-3

Verified step by step guidance

1. Start by simplifying the coefficients: $\frac{16}{12}$. Reduce this fraction to its simplest form.
2. Apply the quotient rule for exponents to the $m$ terms: $m^{-5}$ divided by $m^2$ is $m^{-5-2}=m^{-7}$.
3. Apply the quotient rule for exponents to the $n$ terms: $n^4$ divided by $n^{-3}$ is $n^{4-(-3)}=n^{4+3}=n^7$.
4. Combine the simplified coefficient with the simplified variables: $\frac{4}{3}\cdot m^{-7}\cdot n^7$.
5. Rewrite the expression without negative exponents: $\frac{4n^7}{3m^7}$.

Key Concepts

Exponents and Negative Exponents

Exponents represent repeated multiplication of a base number. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, m^-5 can be rewritten as 1/m^5. Understanding how to manipulate negative exponents is crucial for simplifying expressions without them.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions by canceling common factors in the numerator and denominator. This process often requires factoring polynomials and applying the properties of exponents. The goal is to express the fraction in its simplest form, which is essential for clarity and further calculations.

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves adding or subtracting terms that have the same variable raised to the same power. This process simplifies expressions and makes them easier to work with. In the context of rational expressions, it helps in consolidating terms after applying exponent rules and simplifying fractions.