Problem 51 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ m(m-4) $$

Step-by-Step Solution

Answer

Answer: The simplified expression is m^2 - 4m.

Step 1: Distribute the variable "m"

To begin the process, we will distribute the variable "m" to both the terms inside the parentheses. So the multiplication will look like this: $$ m \cdot m + m \cdot (-4) $$

Step 2: Perform the multiplication

Now that we have distributed the variable "m", we can perform the multiplication: $$ m^2 - 4m $$

Step 3: Combine Like Terms (If Necessary)

In this specific problem, there are no like terms to combine, so our final answer remains the same from Step 2. The simplified expression is: $$ m^2 - 4m $$

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by an expression enclosed in parentheses. To put it simply, you distribute the term outside the parentheses to each term inside.

For example, in the exercise given, we have the expression \( m(m - 4) \). Here's what happens step by step: we take the \( m \) from outside the parentheses and multiply it by each term inside, resulting in \( m \times m \) and \( m \times (-4) \). This gives us \( m^2 \) and \( -4m \), respectively. It's like giving each term inside the bracket a share of the outside term.

Remember, each term inside the parentheses gets multiplied by the term outside. This property holds true for all real numbers, and it's a quick way to expand expressions.

Combining Like Terms

The process of combining like terms is about tidying up an algebraic expression by adding or subtracting terms that have the same variables raised to the same exponents. Think of like terms as matching socks in your drawer; they go together naturally.

In the exercise example, after applying the distributive property, we end up with the terms \( m^2 \) and \( -4m \). If there were any other terms with \( m^2 \) or \( m \) in the expression, we would combine them to simplify our expression further. However, in this case, there are no like terms to combine.

Even so, it is important to always look for like terms after distribution because simplifying an expression makes it easier to understand and work with. Identifying and combining like terms is a bit like simplifying a complex recipe down to basic steps—it makes the result so much clearer!

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the act of reducing an expression to its most basic form without changing its value. This often involves distributing, combining like terms, and cancelling where possible.

In the context of our given exercise, once we have applied the distributive property and looked for like terms, our expression \( m^2 - 4m \) is already in its simplest form. There's nothing more to combine or reduce here. But this step is crucial in more complex expressions, where reducing the expression can involve several rounds of distribution and consolidation of like terms.

Simplification might be the last step, but it's like cleaning up after cooking: it makes the final product—be it an algebraic expression or a delicious meal—neat, tidy, and ready to serve. Always aim for the simplest expression, as it's usually the most useful and understandable form for further mathematical operations or real-world applications.