Simplify each expression. See Example 8. 0.25(8 + 4p) - 0.5(6 + 2p)

Verified step by step guidance

1. Distribute the constants 0.25 and 0.5 to each term inside the parentheses separately. This means multiplying 0.25 by both 8 and 4p, and 0.5 by both 6 and 2p. Write this as: \(0.25 \times 8 + 0.25 \times 4p - (0.5 \times 6 + 0.5 \times 2p)\).
2. Calculate each multiplication individually: multiply 0.25 by 8, 0.25 by 4p, 0.5 by 6, and 0.5 by 2p. Keep the terms separate for now.
3. Rewrite the expression with the results of the multiplications, keeping track of positive and negative signs: \(\text{(result of }0.25 \times 8) + \text{(result of }0.25 \times 4p) - \text{(result of }0.5 \times 6) - \text{(result of }0.5 \times 2p)\).
4. Combine like terms: group the constant terms together and the terms with variable \(p\) together. This means adding or subtracting the constants and the coefficients of \(p\) separately.
5. Write the simplified expression by summing the combined constants and the combined \(p\) terms, resulting in a simplified linear expression.

Key Concepts

Distributive Property

The distributive property allows you to multiply a single term by each term inside a parenthesis. For example, a(b + c) = ab + ac. This property is essential for simplifying expressions like 0.25(8 + 4p) by distributing 0.25 to both 8 and 4p.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. After distributing, terms with 'p' and constant terms should be combined separately to simplify the expression fully.

Decimal Multiplication

Decimal multiplication is the process of multiplying numbers that include decimal points. Understanding how to multiply decimals correctly is important here, as coefficients like 0.25 and 0.5 must be multiplied by terms inside the parentheses accurately.