Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the expression. \(8(3-m)+1(-8)\)

Step-by-Step Solution

Answer

The simplified expression is \(16 - 8m\).

Step 1: Distribute within Parentheses

First, apply the distributive property to the terms inside the parentheses. Multiply 8 by each of the terms in the parentheses: \(3\) and \(-m\). Perform the calculations: \[8 \times 3 - 8 \times m = 24 - 8m\].

Step 2: Simplify Addition and Subtraction

Combine the expression obtained from the distributive step with any other terms outside the parentheses. You have \(24 - 8m + 1(-8)\). Compute \(1 \times -8 = -8\). Now combine all terms: \[24 - 8m - 8\].

Step 3: Combine Like Terms

Identify like terms, which in this case are the constants \(24\) and \(-8\). Combine them:\[24 - 8 = 16\]. So the expression now becomes \[16 - 8m\].

Step 4: Final Answer

Combine all parts from the simplified expression to write down the final simplified result which has no further like terms to combine or simplify. The solution is \(16 - 8m\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations. It's a tool used when you need to multiply a single term by terms inside parentheses. Rather than calculating everything inside the parentheses first, the distributive property lets you handle each term separately.
For example, in the expression \(8(3-m)\), we use the distributive property to multiply 8 by each term inside the parentheses:

• Multiply 8 by 3 to get 24.
• Multiply 8 by \(-m\) to get \(-8m\).

After applying the distributive property, the expression transforms from \(8(3-m)\) to \(24 - 8m\). It's a useful way to simplify and make calculations manageable, especially when dealing with variables. Always remember to multiply each term within the parentheses by the factor outside.

Combining Like Terms

Combining like terms is the process of simplifying expressions by merging terms that have the same variable raised to the same power. This step simplifies expressions by reducing redundant terms.
After applying the distributive property, you may find terms that can be combined, as seen in the expression \(24 - 8m - 8\). Here:

• The numbers 24 and \(-8\) are both constants, so they can be combined.
• The \(-8m\) is not combined with any other term because there's no other term with \(m\).

To combine the like terms, simply add or subtract the constant values, \(24\) and \(-8\), giving you \(16\). This results in a simpler expression: \(16 - 8m\). The process of combining like terms makes algebraic expressions shorter and easier to work with.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This process typically involves applying both the distributive property and combining like terms, as discussed in the previous sections. Simplification means ensuring there are no further calculations or combinations left to perform.
In our solved example, we start with a complex expression: \(8(3-m)+1(-8)\). Following simplification steps:

• Apply the distributive property to remove parentheses, resulting in \(24 - 8m\).
• Combine like terms: \(24\) and \(-8\) to simplify the expression to \(16 - 8m\).

The final result, \(16 - 8m\), is as simplified as possible because no like terms remain to be combined. The goal of algebraic simplification is to make the expression easier to read and work with, minimizing the complexity of both the expression and any further operations you need to perform.