Problem 39 - Algebra 1

Question

List the terms of the expression. $$(-2)(-r)(-r)(-r)$$

Step-by-Step Solution

Answer

The result of multiplying the terms of the expression \(-2(-r)(-r)(-r)\) is \(2r^3\).

Step 1: Understanding the Terms

In this expression, the terms are the factors that are being multiplied together. Here there are four terms: -2, -r, -r, and -r.

Step 2: Combining the Terms

We combine the terms by simply multiplying them together. Since multiplication is associative, it doesn't matter which pairs of terms we multiply together first. In this case, let's start by multiplying the numerical term (-2) with the first -r. This results in \( -2 \times -r = 2r \).

Step 3: Continue Multiplying

Now we multiply 2r from the previous step by the next -r. This results in \( 2r \times -r = -2r^2 \).

Step 4: Finishing Multiplication

Finally, we multiply -2r^2 from the previous step by the last -r. This results in \( -2r^2 \times -r = 2r^3 \).

Key Concepts

Associative Property of Multiplication

When you encounter multiplication in algebra, it's crucial to understand the associative property. This is the rule that tells us that the way in which we group numbers in a multiplication problem does not change the product. In simpler terms, the associative property allows us to regroup factors without altering the result.

Let's look at an example using the given expression from the exercise, \( (-2)(-r)(-r)(-r) \). According to the associative property, you can group the factors in any way. This means that \( (-2) \times (-r) \times (-r) \times (-r) \) can be regrouped as \( ((-2) \times (-r)) \times ((-r) \times (-r)) \) or any other combination, and you would still get the same answer after multiplying. This property is particularly helpful when dealing with multiple terms, as it gives you flexibility to simplify the problem step by step, which can make calculations easier and less error-prone.

Multiplying Negative Numbers

Multiplying negative numbers is often a source of confusion, but it follows a simple rule. When you multiply two negative numbers together, the result is a positive number. This is why, in our exercise, when we multiply negative two, \( -2 \) by negative r, \( -r \) the result is positive 2r. Likewise, if you were to multiply a positive number by a negative number, the result would be negative.

This rule is critical in understanding why \( -2 \times -r = 2r \) and continuing with the steps in our exercise. If we keep multiplying by \( -r \), the sign switches each time we introduce another negative factor. It's almost like playing a game where each negative number changes the 'mood' of the equation. Remember the pattern: negative times negative equals positive, and negative times positive equals negative.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent mathematical relationships. The terms in an algebraic expression are the parts that are added or subtracted. In the case of \( (-2)(-r)(-r)(-r) \), we refer to the individual factors that are multiplied together as terms.

Working with algebraic expressions often requires distributing multiplication over addition or using the aforementioned associative property of multiplication to simplify them. Key to understanding these expressions is recognizing variables, like \( r \) in our exercise, and knowing that they can represent any number. When an algebraic expression includes variables, it might not have a single fixed value, rather it represents a set of possible values, depending on what is substituted for the variable.