Problem 27 - Algebra 2

Question

Name the property illustrated by each statement. If \([3(-2)] z=24,\) then \(-6 z=24\)

Step-by-Step Solution

Answer

Associative Property of Multiplication

Step 1: Identify the operation

First, examine the change from \([3(-2)] z=24\) to \(-6 z=24\). Notice that the expression \(3(-2)\) is replaced by \(-6\). This indicates a simplification of the multiplication expression.

Step 2: Simplify the expression

Calculate \(3(-2)\). Multiply 3 by -2 to get \(-6\). This shows that \(3 \times -2\) simplifies to \(-6\).

Step 3: Recognize the property

The change from \([3(-2)] z=24\) to \(-6 z=24\) illustrates the **Associative Property of Multiplication**. This property states that the way numbers are grouped in multiplication does not affect the product.

Key Concepts

multiplication

Multiplication is one of the fundamental operations in mathematics, used to calculate the total of repeated additions. When you multiply numbers, you essentially add one of the numbers to itself as many times as the other number indicates. For example, multiplying 3 by -2 is the same as adding -2 three times, which gives us -6.
In mathematical terms, multiplication is denoted by symbols such as \(\times\), \(\cdot\), or parentheses. Its significance extends beyond simple arithmetic, forming the backbone of more advanced math concepts and real-world applications, such as calculation of area, scaling, and algebraic expressions.

• **Commutative Property**: The order of numbers does not matter in multiplication. For instance, \(a \times b = b \times a\).
• **Associative Property**: The grouping of numbers does not affect the outcome. This property is crucial in simplifying complex expressions.
• **Multiplicative Identity**: Any number multiplied by 1 remains unchanged.

Understanding these properties can help simplify calculations and solve equations effectively.

simplification of expressions

Simplification of expressions involves reducing an expression to its simplest form. This usually means lowering the complexity or making an expression easier to understand by performing operations like addition, multiplication, or applying algebraic identities.
In expressions involving multiplication, you often combine like terms or use mathematical properties to group and reduce terms. For example, in the expression \(3(-2)z\), simplifying involves performing the multiplication first — which results in \(-6\). This directly demonstrates the use of basic arithmetic operations to streamline expressions, resulting in a cleaner, more direct equation: \(-6z\).

• **Perform Arithmetic Operations**: Carry out any addition, subtraction, multiplication, or division required.
• **Apply Algebraic Identities**: Use identities to factor or expand expressions as needed.
• **Combine Like Terms**: Gather and simplify all similar variables to minimize length.

Mastery in simplification is particularly helpful in solving math problems efficiently and plays an essential role across various math fields including algebra, calculus, and beyond.

mathematical properties

Mathematical properties offer a set of rules that apply to different numerical operations. These properties allow us to manipulate and simplify expressions with confidence. One key property highlighted in this exercise is the **Associative Property of Multiplication**. This property highlights that how we group numbers in multiplication does not impact the result. For instance, when simplifying the expression \([3(-2)] z=24\) to \(-6z=24\), the numbers 3 and -2 are grouped with multiplication to give -6. This simplification into -6 confirms that different groupings or orders in multiplication yield the same number, reflecting on the associative property.

• **Distributive Property**: Multiplies a sum by multiplying each addend separately and then adding the results.
• **Inverse Property**: Multiplying a number by its reciprocal yields 1, acting as a form of "undoing" multiplication.
• **Identity Property**: Multiplying any number by 1 leaves it unchanged, showing the significance of the number 1 in multiplication.

By understanding these properties, one can simplify expressions efficiently and recognize patterns more effectively, which are skills essential not only in mathematics but also in logical reasoning and problem-solving scenarios.