Problem 35 - Elementary Algebra

Question

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations. $$5(6 h+1)$$

Step-by-Step Solution

Answer

Question: Use the commutative property of addition and multiplication to rewrite the expression 5(6h + 1). Answer: (1 + 6h)5

Step 1: First step: Identification

We need to identify the addition and multiplication operations within the expression so we can apply the commutative property. The given expression is $$5(6h + 1)$$. Here, we can see an addition operation (\(6h + 1\)) and a multiplication operation (\(5\) times the result of \(6h + 1\)).

Step 2: Second step: Commutative Property of Addition

Let's first apply the commutative property of addition. In the bracket, we have \(6h + 1\). According to the commutative property of addition, we can change the order of the numbers being added without changing the result. So we can rewrite the expression as $$5(1 + 6h)$$.

Step 3: Third step: Commutative Property of Multiplication

Now let's apply the commutative property of multiplication. Currently, the expression is \(5(1 + 6h)\). If we consider \(5\) as a product of \(5\) and \((1 + 6h)\), we can rewrite the expression as \((\)1 + 6h\()5\). This is because \(ab = ba\) in commutative property of multiplication, so \(5(1 + 6h) = (1 + 6h)5\). So, the final expression after applying the commutative property of addition and multiplication is $$(1 + 6h)5$$.

Key Concepts

Addition

Addition is one of the fundamental operations in mathematics. It combines two or more numbers to get a total sum. In the context of algebra, addition is used to combine like terms and constants.
For instance, in the expression \(6h + 1\), "6h" and "1" are added together. While they are not like terms, we express their relationship using the addition sign.
A crucial property of addition is the commutative property. This property states that the order of the numbers does not affect the sum.

• For example, \(6h + 1\) equals \(1 + 6h\).
• This flexibility assists in rearranging and simplifying expressions without altering their value.

Understanding this property can make manipulation of algebraic expressions easier and more intuitive.

Multiplication

Multiplication is another essential operation in math. It's a process of repeated addition. In algebra, it is often used to distribute numbers over sums and differences.
In expressions like \(5(6h + 1)\), multiplication is represented by the 5 outside the bracket. The idea is that 5 multiplies everything inside the bracket.
The commutative property also applies to multiplication. This means the order in which numbers are multiplied doesn't affect the product.

• In our example, \(5(1 + 6h)\) can be written as \((1 + 6h)5\).
• By using this property, it's possible to reorder and simplify expressions effectively.

Mastery of this property is key to handling complex algebraic problems seamlessly.

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators. They are the language of algebra and allow us to describe relationships and patterns.
For example, in the expression \(6h + 1\), "6h" is a term consisting of the coefficient 6 and the variable "h". The constant 1 is added to it.
Algebraic expressions can be manipulated using fundamental properties of arithmetic operations.

• Terms can be reordered using the commutative properties of addition and multiplication.
• This is crucial for simplifying expressions, solving equations, and modeling real-world situations.

Recognizing different parts of an expression helps in breaking it down and understanding it better.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with solving equations and understanding variables, constants, and algebraic expressions. It's the foundation for all higher-level math topics.
The use of properties like the commutative properties of addition and multiplication is a fundamental skill in algebra.
In this basic level of algebra, you learn to manipulate expressions and solve for unknowns.

• Elementary algebra teaches the skills needed to rearrange equations, making them easier to interpret and solve.
• Concepts such as the commutative property are continually applied to simplify problems and find solutions.

Mastery of elementary algebra is essential for success in more advanced mathematical concepts.