Problem 43 - Elementary Algebra

Question

Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property. $$9 x 2 y$$

Step-by-Step Solution

Answer

Question: Simplify the expression 9xy * 2 using the commutative property of multiplication. Answer: 18xy

Step 1: Identify the elements that can be rearranged

First, we need to identify the elements of the expression that can be rearranged using the commutative property of multiplication. In this case, we have the numbers 9 and 2 as well as the variables x and y.

Step 2: Apply the commutative property of multiplication

Now, we will apply the commutative property to rearrange the elements of the expression. In this case, we can rearrange the factors so that the numbers 9 and 2 come first, followed by the variables x and y, in alphabetical order. So the expression becomes: $$9 \cdot 2 \cdot x \cdot y$$

Step 3: Calculate the product of the numbers

Next, we need to calculate the product of the numbers in the expression, which are 9 and 2. Multiply these numbers together to get: $$18 \cdot x \cdot y$$

Step 4: Write the final simplified expression

Now that we have multiplied the numbers and rearranged the expression using the commutative property of multiplication, we can write the final simplified expression: $$18xy$$

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. They can represent a wide range of relationships and are essential tools in algebra. A simple example is \( 9 \times 2 \times x \times y \) where \(9 \) and \(2\) are numbers, while \(x\) and \(y\) are variables.

Think of variables as 'containers' that can hold different values. The expression is like a recipe that tells you how to combine these ingredients. You \( 'bake' \) your answer by following the operations indicated. When you're asked to simplify an algebraic expression, you're essentially being asked to make the 'recipe' as straightforward as possible without changing the 'flavor' or the value of the original expression.

Simplifying Expressions

Simplifying expressions means to rewrite them in the simplest or most efficient form, making them easier to work with. It often involves combining like terms, using properties of operations, and reducing the number of terms if possible.

This process doesn't change the value of the expression; it just presents it in a cleaner, more digestible form. For our exercise, we use the commutative property, which allows us to move around the multiplication elements. We multiply numbers and variables separately. As a tip, always start by multiplying the constants as we did to get \( 18 \). Then we attach the variables, which in most cases, unless they are like terms, we leave them as they are.

Multiplication of Variables

Multiplication of variables is applying the basic rules of multiplication to letters that represent unknown values. When variables are multiplied, and they are the same, you add their exponents based on the laws of exponents. However, when they differ, such as \(x\) and \(y\), you simply write them side by side after their coefficients.

In the given exercise, we don't have like variables, so we don’t combine exponents but place them next to the multiplied numbers. Remember to maintain alphabetical order, which is a common convention in mathematics, to enhance readability and standardization. Thus, our solution writes \(18\) followed by \(x\) and \(y\), resulting in \(18xy\).