Problem 45 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(5(x+4 m+2)\)

Step-by-Step Solution

Answer

The expression simplifies to \(5x + 20m + 10\).

Step 1: Apply the Distributive Property

The distributive property states that for any real number, \(a(b + c) = ab + ac\). We will apply this property to the expression \(5(x + 4m + 2)\) by multiplying 5 with each term inside the parentheses.

Step 2: Distribute to Each Term

Distribute the 5 to each term inside the parentheses: \(5 \cdot x + 5 \cdot 4m + 5 \cdot 2\). This gives us the expression: \(5x + 20m + 10\).

Step 3: Simplify the Expression

Since we have distributed the 5 to each term and written the expression as \(5x + 20m + 10\), there is no further simplification needed as all terms are unlike terms. This is the simplified version of the expression, written without parentheses.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(x\) or \(m\)), and operation symbols (such as +, -, \, ×, , \, ÷). An algebraic expression represents a quantity, and it can vary depending on the values of the variables it includes.

For instance, in the expression \(x + 4m + 2\), each part of the expression is called a "term." Here, \(x\) is a variable term, \(4m\) is another term where 4 is the coefficient of the variable \(m\), and the 2 is a constant term.

When working with algebraic expressions, it is useful to manipulate them using different properties like the distributive property or combining like terms. This helps in simplifying or solving algebraic equations efficiently. Understanding and correctly applying these techniques is crucial for editing an expression into its simplest form, especially when preparing for further algebraic operations.

Simplification of Expressions

Simplification in algebra involves transforming an expression into its simplest form. This form is the most concise way to write it, with no unnecessary operations or terms. In the context of the given example, after using the distributive property, we arrive at the expression \(5x + 20m + 10\).

To simplify means to look for opportunities to combine like terms or make any arithmetic easier, such as multiplying or distributing, simply put, to make it "as clean as possible."

In the example provided, \(5x\), \(20m\), and \(10\) have no like terms, meaning they cannot be combined further because the variable terms—such as different variables—cannot be combined. This is because 'like terms' must have identical variable parts. Therefore, \(5x + 20m + 10\) is considered fully simplified.

Working with Real Numbers

Real numbers consist of both rational and irrational numbers and include almost anything you can imagine on the number line. These numbers can include positive numbers, negative numbers, and zero, and they can be whole numbers or fractions.

When applying the distributive property in algebraic expressions—as you saw in the example—you are working with real numbers to distribute constants across terms. For example, in the expression \(5(x + 4m + 2)\), 5 is a real number that multiplies the expression inside the parentheses.

• Rational numbers are numbers that can be expressed as fractions, such as \(5\), \(1/2\), or \(-3\).
• Irrational numbers cannot be expressed as a straightforward fraction, like \(\sqrt{2}\) or \( \pi \).

Knowing which are real numbers helps in understanding and solving algebraic expressions because the rules governing real numbers apply to most of the arithmetic we perform, including the distributions like we did in the solution.