Problem 39 - 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. $$(-16)(4)$$

Step-by-Step Solution

Answer

Question: Using the commutative property of addition and multiplication, rewrite the following expression: $$(-16)(4)$$. Answer: $$(4)(-16)$$

Step 1: Understand the Commutative Property

The commutative property states that the order of numbers does not matter in addition or multiplication. In other words, changing the order of numbers will not change the result. For addition, this can be written as: $$a + b = b + a$$ And for multiplication, this can be written as: $$a \times b = b \times a$$

Step 2: Apply the Commutative Property to the Given Expression

We are given the expression $$(-16)(4)$$. We need to apply the commutative property of multiplication to rewrite the expression as an equal number. According to the commutative property: $$(-16) \times (4) = (4) \times (-16)$$

Step 3: Write the Final Answer

Using the commutative property of multiplication, we rewrote the given expression $$(-16)(4)$$ as $$(4)(-16)$$. Both expressions are equal, and we do not need to perform any calculations. The final answer is: $$(4)(-16)$$

Key Concepts

Addition and the Commutative Property

Addition is a fundamental operation of mathematics, and the commutative property is one of its key principles. This property tells us that the order in which we add two numbers does not affect the sum. Whether you add 2 + 3 or 3 + 2, the result remains 5.
This feature makes calculations more flexible and can simplify problem-solving in algebra. It allows us to rearrange numbers to make mental math more manageable. Here’s a quick example:

• Consider adding 9 + 15. You might find it easier to first solve 15 + 9 to quickly arrive at 24 by recognizing familiar number combinations.
  

The commutative property keeps the addition operation consistent regardless of order. This consistency is important in algebra as it ensures that expressions can be manipulated without altering underlying values.

Multiplication and the Commutative Property

Just like addition, multiplication benefits from the commutative property. This means that the product of two numbers will remain the same regardless of their arrangement.
For example, if you have 5 multiplied by 7 or 7 multiplied by 5, the answer will be 35 either way. This property is especially useful when working with more complex expressions.
Applying the commutative property to our example problem, we have:

• Given expression: (-16) \( \times \) 4
• Commutatively rearranged: 4 \( \times \) (-16)

In each case, even though the order of multiplication changes, the result is numerically the same.
This property of multiplication allows flexibility in transforming expressions, making underlying operations simpler to handle, which can be very useful for algebraic manipulations.

Rewriting Expressions Using Commutative Property

The ability to rewrite expressions is crucial in algebra, often achieved using properties like commutativity.
This flexibility allows expressions to be adjusted into forms that might be easier to work with, enabling easier factorization, simplification, or evaluation.
Consider this example involving numbers \(-16\) and \(4\):

• Original expression: \((-16)(4)\)
• Rewritten using the commutative property of multiplication: \((4)(-16)\)

This rewritten form can assist in solving algebra problems by aligning operations to specific problem-solving strategies. It doesn’t change the underlying values; rather it transforms the order for ease of computation.

Solving Algebra Problems with Commutative Property

Algebra problems often require rewriting and simplifying expressions using properties like the commutative property.
This property is vital as it provides algebraic expressions with flexibility to ensure equations remain solvable despite changes in number order.
Consider an equation where rearranging terms efficiently can simplify the path to solving:

• If an expression is \(a + b + c\), and knowing that \(c\) is easily combinable with \(a\), change it to \(a + c + b\).

This strategy ensures calculations are simplified by grouping numbers that are compatible for easy computation. It reduces complexities often encountered when solving longer expressions or equations. Thus, the commutative property serves as a reliable tool in algebra for logically deriving solutions without altering outcomes.