Problem 30 - 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. $$(8+a)(x+6)$$

Step-by-Step Solution

Answer

Question: Rewrite the expression $$(8+a)(x+6)$$ using the commutative property of addition and multiplication. Answer: Two equivalent expressions using the commutative property are $$(a+8)(6+x)$$ and $$(6+x)(a+8)$$

Step 1: Identify the terms in the expression

Given expression: $$(8+a)(x+6)$$ Here the terms to be added are \(8 + a\) and \(x + 6\). We will apply the commutative property of addition to rearrange these terms.

Step 2: Apply commutative property of addition

We can rearrange the terms inside the parentheses as $$a + 8$$ and $$6 + x$$ because addition is commutative. So the new expression becomes: $$(a+8)(6+x)$$

Step 3: Apply commutative property of multiplication

Since multiplication is also commutative, we can rewrite the expression in another equivalent form by changing the order of the factors: $$(6+x)(a+8)$$ The final expressions that are equal to the given expression by applying the commutative property are: $$(a+8)(6+x)$$ and $$(6+x)(a+8)$$

Key Concepts

Commutative Property

The commutative property is a fundamental concept in algebra. It allows us to change the order of numbers and operations while maintaining the same result. For both addition and multiplication, this property states:

• Addition: For any numbers \(a\) and \(b\), \(a + b = b + a\).
• Multiplication: For any numbers \(a\) and \(b\), \(a \times b = b \times a\).

In simple terms, it means you can add or multiply numbers in any order. Therefore, when you see an expression like \((8+a)(x+6)\), you know you can rearrange it. This gives you an option to express it as \((a+8)(6+x)\) or \((6+x)(a+8)\) without changing its value.

Addition

Addition is one of the primary arithmetic operations and is part of the foundation of algebra. It's the process of combining two numbers to get a total sum.
When dealing with expressions, addition behaves similarly. For example, if you have \(8 + a\), adding \(8\) and \(a\) gives you a sum of these terms.
Thanks to the commutative property, these terms can be rearranged as \(a + 8\). This flexibility is beneficial when simplifying complex expressions or solving equations, as observed in the original exercise, where \(8 + a\) becomes \(a + 8\).

Multiplication

Multiplication in algebra involves combining numbers in a way that extends addition. It's about adding a number to itself a specific number of times.

• For instance, \(3 \times 4\) can be viewed as adding \(3\) to itself four times: \(3+3+3+3\).

The commutative property ensures that changing the order of multiplication doesn’t affect the outcome.
In expressions, if we have something like \((8+a) \times (x+6)\), we can rearrange the factors to \((x+6) \times (8+a)\). This property is particularly useful, as it lets us focus on calculus strategies or simplify the problem further.

Equivalent Expressions

Equivalent expressions are different expressions that describe the same quantity. They look different but yield the same value for all variable values whenever you calculate them.

• For instance, \((8+a)(x+6)\) and \((a+8)(6+x)\) are equivalent.
• Even \((6+x)(a+8)\) is another equivalent form.

When solving problems, finding equivalent expressions allows flexibility. It helps in choosing the most convenient form for computations or for simplifying real-world situations.
The use of the commutative property clarifies how rearranging terms and factors can lead to these equivalent expressions, ensuring you always have multiple paths to the same solution.