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Chapter 0: Problem 45

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

Short Answer

Expert verified
-1

Step by step solution

01

- Distribute the constants

Distribute 0.25 to each term inside the first parenthesis and -0.5 to each term inside the second parenthesis. 0.25(8) + 0.25(4p) - 0.5(6) - 0.5(2p)
02

- Simplify the multiplication

Simplify the multiplication inside each term. 0.25(8) = 2 0.25(4p) = p 0.5(6) = 3 0.5(2p) = p. This results in: 2 + p - 3 - p
03

- Combine like terms

Combine the like terms to simplify the expression. 2 - 3 + p - p. This results in: -1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
To simplify algebraic expressions, we often start with the distributive property. This property states that when you multiply a number by a sum, you must distribute the multiplication to each term both inside the parentheses. For example, let's look at the expression: \(0.25(8+4p) - 0.5(6+2p)\). Applying the distributive property, you multiply 0.25 by both 8 and 4p. Similarly, -0.5 is multiplied by 6 and 2p. This results in: \(0.25 \times 8 + 0.25 \times 4p - 0.5 \times 6 - 0.5 \times 2p\). Breaking it down:
  • 0.25 \times 8 = 2
  • 0.25 \times 4p = p
  • -0.5 \times 6 = -3
  • -0.5 \times 2p = -p
Using the distributive property helps break down complex expressions into simpler parts.
Combining Like Terms
After applying the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our example, after applying the distributive property, we had: \(2 + p - 3 - p\). Here, 2 and -3 are like terms since they are constants (numbers without variables). Similarly, \(p\) and \(-p\) are like terms because they both contain the variable \(p\). Combining these:
  • Combine the constants: 2 - 3 = -1
  • Combine the \(p\) terms: \(p - p = 0\)
This results in: \(-1 + 0 = -1\). Combining like terms simplifies the expression significantly.
Basic Algebra
Understanding basic algebra concepts is crucial when simplifying algebraic expressions. Basic algebra involves operations such as addition, subtraction, multiplication, and division, applied to variables and constants. In the given expression, \(0.25(8+4p) - 0.5(6+2p)\), we used basic algebra rules to manipulate and simplify the terms:
  • First, we applied the distributive property.
  • Then, we simplified each term by performing the multiplications.
  • Finally, we combined like terms to get the simplest form: \(-1\).
Key principles of basic algebra include:
1. Distributive Property: Helps break down expressions.
2. Combining Like Terms: Simplifies the expression by merging similar terms.
By mastering these concepts, students can tackle more complex problems with confidence.

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Most popular questions from this chapter

Write each percent as a fraction. Give answers in lowest terms. See Example 13. $$ 47 \% $$

Write each percent as a fraction. Give answers in lowest terms. See Example 13. $$ 2.5 \% $$

Simplify each expression. $$ 15+5 \cdot 2 $$

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ 7 \leq 7 $$

The distributive property can be used to mentally perform calculations. $$\begin{array}{l}38 \cdot 17+38 \cdot 3 \\\=38(17+3) \\\=38(20) \\\=760\end{array} $$Use the distributive property to calculate each value mentally. $$ \frac{8}{5}(17)+\frac{8}{5}(13) $$
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