Problem 22 - Beginning and Intermediate Algebra

Question

Use the commutative and associative properties to simplify each expression. See Example 3 \(\frac{1}{8}(8 z)\)

Step-by-Step Solution

Answer

The expression simplifies to \(z\).

Step 1: Understand the Expression

The given expression is \(\frac{1}{8}(8z)\). Here, we have a fraction \(\frac{1}{8}\) multiplied by the product of \(8\) and \(z\). Our goal is to simplify it using mathematical properties.

Step 2: Apply the Commutative Property of Multiplication

The commutative property of multiplication states that the order in which numbers are multiplied does not change the product. Hence, we can rewrite \(8z\) as \(z \times 8\). So the expression becomes \(\frac{1}{8}(z \times 8)\).

Step 3: Apply the Associative Property of Multiplication

The associative property of multiplication suggests that the way in which numbers are grouped in multiplication does not affect the product. Therefore, we can regroup \(\frac{1}{8}(z \times 8)\) as \((\frac{1}{8} \times 8) \times z\).

Step 4: Simplify the Expression

Calculate \(\frac{1}{8} \times 8\). Since any number multiplied by its reciprocal equals \(1\), \(\frac{1}{8} \times 8 = 1\). Therefore, the expression simplifies to \(1 \times z\), which is just \(z\).

Step 5: Final Simplified Expression

Thus, the expression \(\frac{1}{8}(8z)\) simplifies to \(z\).

Key Concepts

Simplifying Expressions

Simplifying expressions is the process of making mathematical expressions as simple as possible. This often involves combining like terms and making use of mathematical properties. In our example, the expression \( \frac{1}{8}(8z) \) can be simplified using properties of multiplication. Simplification is crucial for solving algebraic problems easily, tracing errors, and making calculations less cumbersome.

The goal of simplifying is to rewrite the expression in the most straightforward form without changing its value. By understanding how each part of the expression works together and applying certain rules, we can reach a simplified form. This becomes especially useful when dealing with complex algebraic equations, providing more clarity and insight into the problem at hand.

Properties of Multiplication

The properties of multiplication are fundamental tools in algebra that allow us to manipulate expressions with ease. Two important properties are the commutative and associative properties.

• Commutative Property: This property states that the order of factors does not affect the product. For example, \( a \times b = b \times a \). In our exercise, we used this property to rearrange \( 8z \) to \( z \times 8 \).
• Associative Property: This property indicates that the grouping of factors does not affect the product. For instance, \( (a \times b) \times c = a \times (b \times c) \). We applied this to group the expression as \( (\frac{1}{8} \times 8) \times z \).

By applying these properties, we can simplify expressions by changing the order and grouping of numbers to find more convenient ways to calculate the result. These strategies are indispensable for solving complex problems and breaking down calculations into more manageable steps.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. They form the language of algebra and are the building blocks for solving equations.

In the given exercise, \( \frac{1}{8}(8z) \) is an algebraic expression involving a fraction and a variable. Each component of the expression works with others to represent a mathematical idea or value. The variables, like \( z \) in this case, are used to represent unknown quantities, while numbers are constants that provide specific values.

When working with algebraic expressions, it's important to understand how different elements interact. Simplifying them helps reveal the underlying structure and main components, making it easier to perform operations and solve equations. The goal of simplifying expressions is not only make them easier to read but also to solve problems efficiently, ensuring that the calculations reflect the correct mathematical relationships.