Problem 39 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (-3.1 u-0.8) 3 $$

Step-by-Step Solution

Answer

The expression without parentheses is \(-9.3u - 2.4\).

Step 1: Recognize the Distributive Property

The distributive property states that \( a(b + c) = ab + ac \). Here, the value to be distributed is 3.

Step 2: Apply the Distributive Property

Multiply each term inside the parentheses by 3: \(-3.1u \times 3 - 0.8 \times 3\).

Step 3: Calculate the result

Evaluate the multiplication: \(-9.3u - 2.4\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics and are used to represent numbers and operations. They consist of variables, numbers, and operation signs (like addition, subtraction, multiplication, and division). When dealing with algebraic expressions, it's important to understand that variables stand in for unknown values that can change depending on the situation.

In our original exercise, the expression \((-3.1u - 0.8)3\) is an algebraic expression. The variable here is \( u \), and the expression shows a multiplication operation applied to the sum of two products. Algebraic expressions allow us to generalize mathematical operations so we can apply them to a variety of scenarios.

Multiplication

Multiplication is one of the four basic operations in arithmetic and is a critical concept in working with algebraic expressions. The operation involves adding a number (the multiplicand) to itself a certain number of times (the multiplier). When handling expressions, multiplication can apply between constants, variables, or a combination of both.

In the provided exercise, multiplication is applied in the expression \((-3.1u - 0.8)3\). This means that each term inside the parentheses will be multiplied by 3. Specifically:

• \(-3.1u imes 3\) results in \(-9.3u\)
• \(-0.8 imes 3\) results in \(-2.4\)

These multiplications break down the more complex expression into simpler parts by distributing the multiplier to each term inside the parentheses.

Simplifying Expressions

Simplifying expressions is about making an algebraic expression as concise as possible while maintaining its original meaning. This process often involves removing unnecessary parentheses, combining like terms, and performing arithmetic operations.

In the original problem, simplifying begins after using the distributive property of multiplication over subtraction in \((-3.1u - 0.8)3\). After distributing, the expression becomes:

• \(-9.3u\)
• \(-2.4\)

Once distributed, the expression does not require further simplification as it is already in its simplest form. This simplification process helps in making the expression more straightforward and easier to work with in subsequent calculations.