Problem 32 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 8(m+7) $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(8m + 56\).

Step 1: Distribute the number outside the parentheses to the terms inside the parentheses

To multiply the expression, we need to distribute the number \(8\) to both the terms inside the parentheses. Distribute \(8\) to \(m\) and \(7\): $$8(m + 7) = 8 \cdot m + 8 \cdot 7$$

Step 2: Perform the multiplications

Now we will perform the multiplication for each term: $$8 \cdot m + 8 \cdot 7 = 8m + 56$$

Step 3: Combine like terms, if any

There are no like terms in the expression \(8m + 56\), so this is the final simplified expression. Final Answer: $$8m + 56$$

Key Concepts

Combining Like Terms

When working with algebraic expressions, it's common to encounter terms that are alike, or 'like terms'. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(3x^2 + 5x^2\), both terms are like terms because they both contain the variable \(x\) raised to the second power. To combine them, simply add their coefficients: \(3+5\) which equals \(8\), making the combined term \(8x^2\).

In our exercise, after distributing the 8, we get the terms \(8m\) and \(56\). These are not like terms because one contains a variable and the other is a constant. Thus, they cannot be combined. Combining like terms is a crucial step when simplifying expressions, as it allows us to make the expression as concise as possible.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(m\) or \(x\)), and operations (such as plus, minus, multiplication, and division). Our exercise features an algebraic expression \(8(m+7)\), which means 8 times the quantity of \(m\) plus 7. It's the job of algebra to find out what the variable (in this case, \(m\)) equals, but before that, we usually need to simplify the expression.

To work with algebraic expressions effectively, understanding and applying the correct order of operations is essential. In the given exercise, we first distribute, then multiply, and as the final step, we combine like terms if there are any. Algebraic expressions are the cornerstone of algebra and are fundamental to understanding more complex mathematical concepts.

Simplifying Expressions

Simplifying expressions is a process by which we reduce an algebraic expression to its most basic form. This often involves several steps, including distributing multiplication over addition or subtraction (known as the distributive property), combining like terms, and canceling operations. The goal is to make the expression easier to understand and work with.

The exercise \(8(m+7)\) is a prime example of this process. First, we apply the distributive property to eliminate the parentheses by multiplying 8 by both \(m\) and 7, resulting in the expression \(8m + 56\). Since this expression contains no like terms, it is already in its simplest form. Understanding the steps to simplify expressions is invaluable for solving algebraic equations and building a foundation for higher-level math.