Problem 78 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(10[0.5(8-2 z)]\)

Step-by-Step Solution

Answer

Simplified expression: \(40 - 10z\)

Step 1: Simplify Inside Parentheses

Start by simplifying the expression inside the parentheses. \[ 8 - 2z \] Leave it as it is, since no specific values are given for \(z\).

Step 2: Apply the 0.5 Factor

Next, apply the 0.5 factor to each term inside the parentheses. Distribute 0.5 across \(8 - 2z\).\[ 0.5 \times 8 - 0.5 \times 2z \] This simplifies to \[ 4 - z \]

Step 3: Apply the Outer Factor

Finally, multiply the simplified expression by 10.\[ 10 \times (4 - z) \]Distribute 10 across the terms inside the parentheses:\[ 10 \times 4 - 10 \times z \]This yields \[ 40 - 10z \]

Key Concepts

Understanding Arithmetic Operations

Arithmetic operations are basic mathematical procedures like addition, subtraction, multiplication, and division. These operations are the building blocks of algebra and are essential for simplifying expressions.
In the given exercise, you can see several arithmetic operations:

• First, the subtraction inside the parentheses: \(8 - 2z\).
• Then, the multiplication by 0.5: \(0.5 \times 8 - 0.5 \times 2z\).
• Finally, the multiplication by 10: \(10 \times (4 - z)\).

Each step involves performing these arithmetic operations correctly.
Breaking down each step helps in understanding how expressions transform and simplify.

Using the Distributive Property

The distributive property is a fundamental concept in algebra. It allows you to remove parentheses by distributing the multiplication over addition or subtraction inside the parentheses.
In mathematical terms, it states that \(a(b + c) = ab + ac\).
In the exercise, the distributive property is used twice:

• First, distribute 0.5 to both terms inside \(8 - 2z\): \(0.5 \times 8 - 0.5 \times 2z\), which simplifies to \(4 - z\).
• Second, distribute 10 to both terms inside \(4 - z\): \(10 \times 4 - 10 \times z\), resulting in \(40 - 10z\).

Understanding how to apply the distributive property makes it easier to simplify more complex algebraic expressions.

What Are Algebraic Expressions?

Algebraic expressions are combinations of numbers, variables (like z), and arithmetic operations. They do not have an equal sign, unlike algebraic equations.
Here are some key points to remember:

• Variables represent unknown values.
• You can simplify expressions by combining like terms and using arithmetic operations.
• Distributing and factoring are common techniques in manipulating algebraic expressions.

In the given problem, \(10[0.5(8-2z)]\) is an algebraic expression that needs to be simplified.
Following the correct order of operations is crucial: simplifying inside parentheses first, then applying multiplication factors.
The result, \(40 - 10z\), is a simplified form of the initial expression.