Simplify each expression. See Example 1. (½ mn) (8m²n²)

Verified step by step guidance

1. Identify the expression to simplify: \((\frac{1}{2} mn)(8m^{2}n^{2})\).
2. Apply the associative property of multiplication to group the coefficients and variables separately: \((\frac{1}{2} \times 8) \times (m \times m^{2}) \times (n \times n^{2})\).
3. Multiply the numerical coefficients: \(\frac{1}{2} \times 8\).
4. Use the laws of exponents to combine like bases: \(m^{1} \times m^{2} = m^{1+2} = m^{3}\) and \(n^{1} \times n^{2} = n^{1+2} = n^{3}\).
5. Write the simplified expression by combining the results from the previous steps: (result of coefficient multiplication) \(m^{3} n^{3}\).

Key Concepts

Multiplication of Algebraic Expressions

This involves multiplying coefficients (numerical parts) and variables separately. When multiplying variables with exponents, you add the exponents if the bases are the same. For example, m × m² = m³.

Properties of Exponents

When multiplying powers with the same base, add their exponents: a^m × a^n = a^(m+n). This rule helps simplify expressions like m × m² into m³, making it easier to combine terms.

Simplifying Numerical Coefficients

Multiply the numerical coefficients separately from the variables. For example, (1/2) × 8 = 4. Simplifying coefficients first helps reduce the expression before combining variable terms.