Problem 20 - Beginning and Intermediate Algebra

Question

Use the commutative and associative properties to simplify each expression. See Example 3 \(2(42 x)\)

Step-by-Step Solution

Answer

The expression simplifies to \(84x\).

Step 1: Apply the Commutative Property

The commutative property allows us to change the order of multiplication without affecting the product. In this case, we rewrite the expression by swapping 2 and 42. So, \(2(42x) = (2 \cdot 42)x\).

Step 2: Perform the Multiplication Inside

Next, we perform the multiplication inside the parentheses. Multiply 2 and 42 to get \(84\). Thus, \((2 \cdot 42)x = 84x\).

Key Concepts

Associative Property

The associative property is a fundamental concept in mathematics that deals with grouping. It states that when three or more numbers are multiplied or added, the way the numbers are grouped does not change the result. This property can be immensely helpful in simplifying algebraic expressions.

For multiplication, the associative property is expressed as:

• \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)

This means that regardless of how you group the numbers or variables within parentheses, the result remains unchanged. By leveraging this property, you can rearrange and group terms in algebraic expressions to make calculations more straightforward.

In the context of simplifying expressions like our example, the associative property allows rearrangement of multiplication which can often make calculations mentally easier, especially when dealing with variables and constants. However, keep in mind that the associative property only works for addition and multiplication, not subtraction or division.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are the building blocks in algebra, representing quantities and operations. An algebraic expression could be as simple as \( x + 2 \) or more complex, like \( 3x^2 - 2xy + y^2 \).

Key components of algebraic expressions include:

• **Variables:** Symbols (often letters) that represent unknown or variable quantities.
• **Constants:** Fixed values that don't change.
• **Coefficients:** Numbers multiplied by the variables.
• **Operators:** Such as addition (+), subtraction (-), multiplication (\( \cdot \)), and division (\( \div \)).

Algebraic expressions allow us to generalize mathematical concepts and work with variables instead of specific numbers. When simplifying these expressions, understanding the role of each component is crucial. For instance, in the expression \(84x\), 84 is the coefficient of the variable \(x\). By simplifying such expressions through properties like the commutative and associative, we can make calculations much simpler and more efficient.

Simplification Techniques

Simplification techniques are crucial in making sense of complex algebraic expressions by reducing them into simpler forms. These techniques often involve using properties of numbers, such as the commutative and associative properties, to rearrange and combine like terms.

Here are some key techniques:

• **Reorder Terms:** Use the commutative property to change the order of terms or factors to make simplification easier.
• **Regroup Factors:** Apply the associative property to regroup terms or factors such that computations become more manageable.
• **Combine Like Terms:** Merge terms that have the same variables raised to the same power, which helps in reducing the complexity of the expression.

In our example with \(2(42x)\), simplification involved first using the commutative property to rearrange factors, then performing the multiplication inside the parenthesis to reach a more straightforward result, \(84x\).

By systematically applying these techniques, you can handle a wide range of algebraic expressions, making problem-solving more accessible and efficient.