Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. 16m^-5n⁴/12m²n^-3

Verified step by step guidance

1. Start by writing the given expression clearly: \(\frac{16m^{-5}n^{4}}{12m^{2}n^{-3}}\).
2. Simplify the coefficients (numerical part) by dividing 16 by 12. This can be written as \(\frac{16}{12}\), which can be reduced to its simplest form.
3. Apply the quotient rule for exponents to the variable \(m\): when dividing like bases, subtract the exponents. So, \(m^{-5} \div m^{2} = m^{-5 - 2} = m^{-7}\).
4. Apply the quotient rule for exponents to the variable \(n\): \(n^{4} \div n^{-3} = n^{4 - (-3)} = n^{4 + 3} = n^{7}\).
5. Rewrite the expression with the simplified coefficient and variables, and then eliminate any negative exponents by rewriting \(m^{-7}\) as \(\frac{1}{m^{7}}\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include dividing powers with the same base by subtracting exponents and multiplying powers by adding exponents. For example, \(a^m / a^n = a^{m-n}\). These rules help simplify expressions like the given problem.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions by factoring and canceling common factors in the numerator and denominator. This process also applies to variables with exponents, allowing the expression to be written in simplest form without negative exponents.

Eliminating Negative Exponents

Negative exponents indicate reciprocals, such as \(a^{-n} = 1/a^n\). To write answers without negative exponents, rewrite terms with negative powers as positive exponents in the denominator or numerator accordingly. This ensures the expression is in standard simplified form.