Problem 94 - Elementary Algebra Within Reach

Question

Use the Distributive Property to expand the expression. $$-z\left(x z-2 y^{2}\right)$$

Step-by-Step Solution

Answer

The result of the expression after applying the distributive property and simplifying is: \( -z^2x + 2zy^2 \)

Step 1: Identify the Terms

Identify the values or states that are to be distributed. Here, \( -z \) is multiplied with \( (xz - 2y^2) \), thus \( -z \) will be distributed to both \( xz \) and \( -2y^2 \).

Step 2: Apply the Distributive Property

Multiply \( -z \) with \( xz \) and \( -2y^2 \) separately. As a result, you obtain \( -z \times xz \) and \( -z \times -2y^2 \).

Step 3: Simplify the Result

Simplify the results of multiplication to get the final answer, which leads to \( -z^2x \) and \( 2z y^2 \).

Key Concepts

Expanding Expressions Using the Distributive Property

Expanding expressions is all about opening up the brackets to reveal what's inside. This process helps us simplify and solve algebraic expressions more easily. The distributive property is a key tool here, allowing you to multiply each term inside a bracket by a factor outside of it. In our example, the task is to expand the expression

• \(-z(xz - 2y^2)\)

The distributive property will guide us as we distribute the \(-z\) across both terms inside the bracket. Think of it like opening a gift to see what's inside. By following this process, we get new expressions that are easier to work with or simplify in further calculations. This is particularly useful in algebra, where simplifying expressions is often a crucial step.

The key steps include:

• Identifying each term inside the brackets.
• Applying the multiplication to each of those terms.
• Simplifying the results for a cleaner solution.

Understanding Algebraic Expressions

Algebraic expressions contain numbers, variables, and operations. These combinations form the basis for much of algebra. The original expression

• \(-z(xz - 2y^2)\)

is a perfect example. It has variables \(z\), \(x\), and \(y\), coupled with numbers and operations like subtraction and multiplication.

Variables like \(z\), \(x\), and \(y\) can often change, which is why they are critical in forming expressions that can solve a wide array of problems. Understanding the role of each part of the expression is crucial. They form the building blocks with which you can explore different values and equations in math.

Being adept with algebraic expressions also helps in understanding functions and equations, making them an essential tool in both simple and complex problems.

Applying Mathematical Operations

To work efficiently with mathematical expressions, it's important to grasp the core operations like addition, subtraction, multiplication, and division. In this problem, multiplication is the main operation, aided by the distributive property. By applying multiplication across all terms, we simplify our expression

• Start with: \(-z(xz - 2y^2)\)
• Expand to: \(-z \times xz\) and \(-z \times -2y^2\)
• Simplify to: \(-z^2x + 2zy^2\)

Multiplying \(-z\) with \(xz\) results in \(-z^2x\). Similarly, multiply \(-z\) with \(-2y^2\) to achieve \(2zy^2\). Notice how the negative signs interact to change the outcome.

Each operation transforms the expression step by step, unraveling a more straightforward form. Remember, practicing these operations regularly boosts your confidence and skill in solving mathematical problems efficiently.