Problem 53 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$(u-w)(-8)$$

Step-by-Step Solution

Answer

The equivalent expression is \(-8u + 8w\).

Step 1: Understanding the Distributive Property

The Distributive Property allows us to multiply each term inside a parenthesis by a factor outside. For an expression \[a(b + c) = ab + ac\] it means multiplying \(a\) with each of the terms inside the parentheses \(b\) and \(c\). In this exercise, we will apply the Distributive Property to the expression \((u-w)(-8)\).

Step 2: Applying the Distributive Property

According to the Distributive Property, multiply \(-8\) with each term inside the parentheses separately. So, \[ (u-w)(-8) = -8(u) + (-8)(-w) \] which breaks down to multiplying \(-8\) with \(u\) and then \(-8\) with \(-w\).

Step 3: Calculating Each Term

Perform each multiplication separately. For the first term: \[-8(u) = -8u\] For the second term: \[-8(-w) = 8w\] (Note the double negative becomes positive for \(-8)(-w)\).

Step 4: Writing the Equivalent Expression

Combine the calculated terms to write the equivalent expression: \[-8u + 8w\]This is the expression equivalent to \((u-w)(-8)\) using the Distributive Property.

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operations. In this exercise, the expression

• \((u-w)(-8)\) is an algebraic expression involving a subtraction between the variables \(u\) and \(w\) inside parentheses, multiplied by the integer \(-8\) outside.

Each part of the expression is referred to as a "term," and terms are typically linked by an addition or subtraction operator. In this expression, the term \((u-w)\) is enclosed in parentheses. This means we should treat it as a single unit during operations like multiplication.
Managing expressions requires an understanding of how variables and constants behave during operations, like when using the distributive property to simplify expressions.

Multiplication of Integers

Multiplication is one of the four basic operations in math, and it applies to integers just like any other number. In this case, we need to multiply an integer with each part of an algebraic expression.

• First, observe the integer outside the parentheses, which in this example is \(-8\).

We’ll multiply \(-8\) with each term inside the parentheses:

• Multiply \(-8\) with \(u\) to get \(-8u\).
• Multiply \(-8\) with \(-w\) to get \(8w\). The multiplication of two negative numbers results in a positive number.

Each multiplication involves simply applying basic rules of multiplication, especially focusing on the sign (positive or negative) of integers being multiplied.

Properties of Operations

Properties of operations are the rules that govern how numbers and expressions can be manipulated. The prominent one in this case is the distributive property.
The distributive property allows us to multiply a single term by every term inside a set of parentheses. This simplifies complex expressions and is stated as follows:

• For any numbers or variables \(a\), \(b\), and \(c\): \(a(b + c) = ab + ac\).

Applying this property to the expression \((u-w)(-8)\) involves multiplying \(-8\) by each term inside the parentheses:

• Multiply \(-8\) with \(u\), producing \(-8u\).
• Multiply \(-8\) with \(-w\), resulting in \(+8w\).

Finally, we combine these results to provide the simplified expression \(-8u + 8w\).
This property makes calculations with algebraic expressions easier and is fundamental for solving equations and further simplifications.