Problem 10 - Algebra for College Students

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$[(-7)(4)](-25)=(-7)[4(-25)]$$

Step-by-Step Solution

Answer

Associative property of multiplication.

Step 1: Identify the Property

Look at the given expression: \([(-7)(4)](-25)=(-7)[4(-25)]\). Notice how the order of the multiplication in the brackets changes. Initially, \(4\) is multiplied by \((-25)\) and later, \(-7\) is multiplied by the result of that product.

Step 2: Identify the Associative Property

Recognize that the associative property of multiplication allows numbers to be grouped in different ways without changing the result. The expression shows a change in grouping from \([(-7)(4)](-25)\) to \((-7)[4(-25)]\), which is consistent with the associative property of multiplication: \((ab)c = a(bc)\).

Key Concepts

Multiplication

Multiplication is one of the four elementary arithmetic operations that involves combining numbers to get a total amount. If you have a multiplication problem like \(a \times b\), it means you are combining \(a\) groups of \(b\) together.

Here are some important points to remember about multiplication:

• It allows you to add equal groups quickly. For example, \(5 \times 3\) is the same as adding \(5\) three times.
• The result of multiplication is called a product.

Multiplication is a crucial operation in mathematics, widely used in various calculations, from simple arithmetic to more advanced problems. Understanding how numbers interact through multiplication helps in solving equations, understanding patterns, and exploring mathematical relationships.

Commutative Property

The commutative property is a fundamental principle in mathematics that applies to both addition and multiplication. This property states that the order in which you add or multiply numbers does not affect their sum or product.

Let's think about multiplication and how it uses this property:

• For example, with numbers \(a\) and \(b\), the commutative property tells us \(a \times b = b \times a\).
• This means if you multiply \(3 \times 4\), you will get the same result as \(4 \times 3\).

This property is very helpful in simplifying expressions and calculations.
By rearranging numbers using this property, you can often make problems easier to understand and solve.

Properties of Operations

In mathematics, properties of operations refer to a set of rules that allow us to understand how numbers behave under various operations like addition, subtraction, multiplication, and division. These properties help to simplify calculations and solve equations efficiently.

Some of the key properties associated with operations include:

• Associative Property: This allows you to change the grouping of numbers, especially in addition and multiplication, without affecting the outcome. For example, \((a+b)+c = a+(b+c)\) or \((ab)c = a(bc)\).
• Distributive Property: This connects addition and multiplication, such as \(a(b+c) = ab + ac\).
• Identity Property: This ensures that adding zero or multiplying by one doesn't change the number, such as \(a + 0 = a\) or \(a \times 1 = a\).

Understanding these properties lets you manipulate and approach mathematical problems confidently, knowing that these steadfast rules always apply.