Problem 51 - Elementary Algebra

Question

Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property. $$1(x+2 y)(6+z) 9(3 x+5 y)$$

Step-by-Step Solution

Answer

Question: Using the commutative property of multiplication, simplify the expression 1(x+2y)(6+z)9(3x+5y). Answer: 9(x+2y)(6+z)(3x+5y)

Step 1: Identify the commutative property of multiplication

The commutative property of multiplication means that, for any values a, b, and c, we can rearrange the order in which we multiply, and the result will be the same. For example, $$(a \cdot b) \cdot c = a \cdot (b \cdot c) = (b \cdot c) \cdot a$$.

Step 2: Reorder the expression

Let's reorder the given expression using the commutative property of multiplication: $$1(x+2 y)(6+z) 9(3 x+5 y) = 1 \cdot 9 \cdot (x+2 y) \cdot (6+z) \cdot (3 x+5 y)$$.

Step 3: Simplify the expression

Now, we can further simplify the expression by combining the constant values and grouping the terms with variables: $$1 \cdot 9 \cdot (x+2 y) \cdot (6+z) \cdot (3 x+5 y) = 9(x+2y)(6+z)(3x+5y)$$. So, the simplified expression is: $$9(x+2y)(6+z)(3x+5y)$$.

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations that we use all the time in math. It involves finding the total of one number repeated a certain number of times. When we multiply two numbers together, we rely on a few properties to make calculations easier.
One crucial property is the **commutative property of multiplication**. This property states that the order of numbers doesn't affect their product. For instance, if you multiply two numbers, say 4 and 5, the result will be the same whether you do 4 times 5 or 5 times 4.

• This concept allows us to rearrange numbers within an expression to simplify problems.
• In algebra, this property can be especially handy when dealing with complex expressions.
• It helps in grouping terms in a way that makes the calculation process easier or more efficient.

By understanding the commutative property, students can solve multiplication problems more flexibly.

Simplification

Simplification in mathematics involves reducing expressions to their simplest form. This often means condensing longer expressions into shorter, more manageable ones.
In our problem, simplification is carried out by applying the commutative property of multiplication. We rearrange and combine like terms and constant values, making the overall expression shorter and easier to understand.
Effective simplification involves:

• Identifying patterns or properties that can make the process easier, like the commutative property.
• Reordering expressions to match these patterns.
• Consolidating numbers and like terms wherever possible.

Simplifying saves time and helps to see relationships between different parts of an expression, leading to a better grasp of the problem's structure.

Expressions

In mathematics, an expression is a combination of numbers, variables, and operators (like addition or multiplication). Expressions can represent numbers in a flexible way and can be simplified or evaluated in different contexts.
The expression given in the exercise involves both numbers and variables, and it shows how these can be manipulated and transformed using properties such as the commutative property of multiplication.
Important aspects of expressions include:

• The ability to rearrange them using mathematical operations.
• Simplifying them using properties and rules such as order of operations or commutativity.
• Interpreting them correctly, which requires understanding what each symbol and operation stands for.

Mastery of expressions is fundamental to progressing in algebra and other areas of mathematics, as they form the building blocks for more complex equations and functions.

Constant Values

Constant values in mathematics are numbers that do not change. These are often combined with variables in expressions. In the solution provided, the number 9 is a constant value. It stands alone without any variables attached.
Here's why constant values are important:

• They serve as the unchanging parts of an expression, giving it stability.
• They can be manipulated just like other terms in the expression. For instance, constants can be factored out or combined with other constants to simplify expressions.
• Understanding how to handle constants is crucial, as they often play a role in defining the solution to an expression or equation.

In the given problem, understanding the role and manipulation of the constant value helped us simplify the expression from complex to manageable.