Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 16m^-5n⁴/12m²n^-3

Verified step by step guidance

1. Start by writing the expression clearly: \(\frac{16m^{-5}n^{4}}{12m^{2}n^{-3}}\).
2. Simplify the coefficients (numerical parts) by dividing 16 by 12. This gives \(\frac{16}{12}\), which can be reduced to its simplest form.
3. Apply the quotient rule for exponents to the variables with the same base. For \(m\), subtract the exponent in the denominator from the exponent in the numerator: \(m^{-5 - 2}\). For \(n\), do the same: \(n^{4 - (-3)}\).
4. Rewrite the expression with the simplified coefficient and the new exponents: \(\text{(simplified coefficient)} \times m^{\text{new exponent}} \times n^{\text{new exponent}}\).
5. Finally, rewrite any negative exponents as positive exponents by moving the base to the opposite part of the fraction (numerator to denominator or vice versa) to ensure the answer has no negative exponents.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers, such as multiplying powers with the same base by adding exponents, dividing by subtracting exponents, and handling negative exponents by rewriting them as reciprocals. These rules are essential for simplifying algebraic expressions correctly.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions by factoring numerators and denominators, canceling common factors, and applying exponent rules. This process helps to write expressions in their simplest form, making them easier to interpret and use.

Eliminating Negative Exponents

Negative exponents indicate reciprocals, so to write answers without negative exponents, rewrite terms with negative powers as fractions with positive exponents. This step ensures the expression is presented in a standard, simplified form preferred in algebra.