Problem 24 - Intermediate Algebra

Question

Complete each statement so that the indicated property is illustrated. See Example 1. \(2(5 \cdot 97)=\) ______ Associative property of multiplication

Step-by-Step Solution

Answer

\((2 \cdot 5) \cdot 97\) illustrates the associative property.

Step 1: Understand the associative property

The associative property of multiplication states that the way you group numbers in a multiplication problem does not change the product. Therefore, for any numbers \(a\), \(b\), and \(c\), \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).

Step 2: Identify the numbers involved

In the expression \(2(5 \cdot 97)\), the numbers involved are \(2\), \(5\), and \(97\).

Step 3: Apply the associative property

Using the associative property, we can regroup the numbers without changing the order. So, \(2(5 \cdot 97)\) becomes \((2 \cdot 5) \cdot 97\).

Step 4: Complete the statement

The statement illustrating the associative property is \((2 \cdot 5) \cdot 97\), which shows that the grouping has been changed while maintaining the same numbers in the same order.

Key Concepts

Multiplicative Properties

Understanding multiplicative properties is essential for mastering mathematical operations. One of the key multiplicative properties is the associative property. This property tells us that when we multiply three or more numbers, the manner in which the numbers are grouped does not affect their final product. For instance, if you consider numbers \(a\), \(b\), and \(c\), then the associative property states that \((a \cdot b) \cdot c = a \cdot (b \cdot c)\). This can be quite handy when solving problems, as it allows for flexibility in computation. It simplifies calculations by enabling us to rearrange terms, which can make mental math easier.

Key features of the associative property include:

• Only applies to multiplication and addition, not to subtraction or division.
• Helps in simplifying complex arithmetic expressions.

It is important to remember that although the associative property allows us to change groupings, it does not allow us to change the order of the numbers, which is a separate property known as the commutative property.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent a value or a series of values. In the context of the exercise, the expression \(2(5 \cdot 97)\) is an example of an algebraic expression. Here, numbers are combined using multiplication, but in general, algebraic expressions can also include addition, subtraction, and division.

To better understand algebraic expressions, consider their basic components:

• **Numbers:** Known as constants when they do not change.
• **Variables:** Letters like \(x\) or \(y\) that represent unknown numbers or values.
• **Operators:** Signs like +, -, \(\cdot\), and ÷ that indicate operations.

Recognizing and correctly interpreting algebraic expressions is crucial for solving various math problems. The ability to apply properties like associativity is just one aspect of working with algebraic expressions. These properties help in both simplifying expressions and solving equations efficiently.

Mathematical Operations

Mathematical operations form the basis of all arithmetic and algebra. These operations include addition, subtraction, multiplication, and division. They are the building blocks of number manipulation. In the problem concerning the associative property, multiplication is the focus, and understanding how it interacts with associative rules is vital.

Here's a deeper look into multiplication as a mathematical operation:

• **Defined as repeated addition:** Multiplying two numbers is like adding one number to itself multiple times.
• **Has properties like associative, commutative, and distributive:** These properties make calculations more flexible.

Using operations effectively requires not only knowledge of their fundamental characteristics but also how they can be combined. When you become comfortable with the basic operations, you can easily explore more advanced topics like distributions and factorization. Consistent practice with these operations, especially using properties like associative property, leads to better problem-solving skills and mathematical fluency.