Problem 47 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(-4(1-2 m+n)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-4 + 8m - 4n\).

Step 1: Apply the Distributive Property

To remove the parentheses, apply the distributive property by multiplying \(-4\) with each term inside the parentheses: \(-4\times 1\), \(-4\times(-2m)\), and \(-4\times n\).

Step 2: Simplify Each Term

Compute each term from Step 1:\(-4\times 1 = -4\)\(-4\times(-2m) = 8m\) (since multiplying two negative numbers gives a positive number)\(-4\times n = -4n\).

Step 3: Combine Terms

Now, combine all the terms obtained after distribution: \(-4 + 8m - 4n\). No further simplification is possible here as all terms are unlike.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. In these expressions, the variables represent unknown values and play a key role in algebra. Understanding algebraic expressions is vital because they are the foundation of many mathematical problem-solving processes.

Consider an expression within parentheses, like \(1 - 2m + n\), which includes numbers (constants) and variables. Each part of this expression, such as \(1\), \(-2m\), and \(n\), is called a 'term'. Terms in algebraic expressions are separated by addition or subtraction operators.

When approaching these expressions, it's often necessary to simplify or transform them using mathematical properties such as the distributive property. This step helps make the expression easier to work with or solve for a particular variable.

Simplification Process

The simplification process involves converting a complex expression into a simpler or more manageable form. It's an essential step in solving algebraic problems and requires careful attention to detail.

To simplify the expression using the distributive property, apply the rule that states \(a(b + c) = ab + ac\), which means multiplying each term inside the parentheses by the factor outside. The main goal is to remove the parentheses and reduce the expression to basic terms.

Start with our specific expression: \-4(1 - 2m + n)\. The steps include:

• Multiply \(-4\) by each term within the parentheses. So, compute \-4 \times 1\, \-4 \times (-2m)\, and \-4 \times n\.
• Simplify these calculations. In this case: \-4\, \8m\, and \-4n\.
• Combine the simplified terms into a final expression: \-4 + 8m - 4n\.

It's important to check if any like terms exist to further combine, but as in this example, each term remains distinct.

Negative Multiplication

Negative multiplication is a crucial arithmetic operation, especially in the simplification of expressions. It deals with multiplying negative numbers or multiplying negative with positive numbers. Understanding this concept ensures accurate calculations in algebra.

Here, the exercise involves multiplying \(-4\) by each element inside the parentheses \(1 - 2m + n\). When multiplying:

• A negative number by a positive number, the result is always negative. Hence, \-4 \times 1 = -4\.
• A negative number by a negative number, the result is always positive. So, \-4 \times (-2m) = 8m\.
• A negative number by a positive variable yields a negative result, giving \-4 \times n = -4n\.

These operations reflect the rules for multiplying numbers with different signs, ensuring each step aligns with fundamental math principles. Understanding and applying these rules accurately simplifies expressions effectively, avoiding common pitfalls in algebraic manipulation.