Problem 64 - Elementary Algebra

Question

Simplify the following problems using the commutative property of multiplication. You need not use the distributive property. $$ 8 x 3 y $$

Step-by-Step Solution

Answer

Question: Simplify the given expression using the commutative property of multiplication: $8x3y$ Answer: $24xy$

Step 1: Identify the multiplication terms

Identify the multiplication terms in the given expression: $$ 8x3y $$ The multiplication terms are 8, 3, x, and y.

Step 2: Apply the commutative property

Using the commutative property of multiplication, we can rearrange the multiplication terms: $$ 8x3y = 8\times 3\times x\times y $$

Step 3: Rearrange the terms

Rearrange the terms to group the constants and the variables together: $$ 8\times 3\times x\times y = (8\times3) \times (x\times y) $$

Step 4: Multiply the constants and the variables

Multiply the constants together and the variables together: $$ (8\times3)\times(x\times y) = 24xy $$ So, the simplified expression is: $$ 24xy $$

Key Concepts

Simplifying Algebraic Expressions

Understanding how to simplify algebraic expressions is a cornerstone of mastering elementary algebra. Simplifying involves reducing expressions to their most basic form while maintaining their original value. A simplified expression is easier to read and often makes subsequent algebraic operations less complicated.

Take, for example, an algebraic expression involving multiplication such as \( 8 \times 3y \). To simplify this expression, we look at each part—namely the coefficients (numerical factors) and the variables (letters representing unknown values). Our goal here is to combine like terms and apply the commutative property of multiplication to organize and reduce the expression.

Identifying Like Terms

Like terms are terms that have the same variable part. In our example, since there is only one variable 'y', all constants can be multiplied together as one term, and the variable 'y' as another. Simplifying involves combining these constants before attaching the variable. This process leads us to an expression that is easier to work with and understand for further algebraic manipulation.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the rules and operations for manipulating algebraic expressions and solving for unknowns within equations. It serves as the foundation for more advanced mathematical studies and practical problem-solving in various fields.

In the context of our exercise, elementary algebra teaches us how we can rearrange and combine numbers and variables. It's not just about finding a solution—it's about understanding the relationship between numbers, variables, and operations. The exercise provided showcases just one aspect of elementary algebra: the use of the commutative property to rearrange and simplify multiplication expressions without altering the product's value.

The Importance of Properties

It's vital to know different properties of operations, such as commutative, associative, and distributive properties because they are tools that allow algebraists to reshape expressions into forms that are more useful or easier to understand. With a firm grasp of elementary algebra, you can simplify expressions efficiently, solve for variables, and ultimately learn to craft and manipulate mathematical models that represent real-world scenarios.

Properties of Multiplication

Properties of multiplication are rules that apply to the multiplication of numbers and variables. These properties help us understand how to manipulate and simplify expressions in a consistent and logical way. One of the fundamental properties we use is the commutative property of multiplication.

The commutative property states that the order in which two numbers or terms are multiplied does not affect the product. In algebraic terms, this means that \( a \times b = b \times a \). This property is particularly useful when simplifying expressions because it allows us to group numbers (coefficients) and variables together in a way that makes them easier to manage. In the exercise, by applying this property, we group 8 and 3 together to multiply them, resulting in 24, making the final expression \( 24xy \), which is the product of the coefficients and variable neatly multiplied.

Applying the Property

Remembering that the order of multiplication doesn't change the outcome provides the flexibility to rearrange terms for easier computation. For example, in mental math, multiplying 8 by 3 might be more straightforward than multiplying 3 by 8 for some individuals. Knowing that you can flip the terms without changing the product can simplify the arithmetic. It's a simple yet powerful tool in algebra that can greatly streamline the simplification process.