Problem 2 - Algebra and Trigonometry

Question

Complete each statement, and name the property of real numbers you have used. (a) \(a b=\)_______ ; ______ Property (b) \(a+(b+c)=\)_______; ______Property (c) \(a(b+c)=\)_______ ; ______Property

Step-by-Step Solution

Answer

(a) \(ab = b \cdot a\); Commutative Property (b) \(a + (b + c) = (a + b) + c\); Associative Property (c) \(a(b + c) = ab + ac\); Distributive Property

Step 1: Analyzing Part (a)

The expression in part (a) is \(a \cdot b\). According to the Commutative Property of Multiplication, which states that the order in which you multiply numbers does not change the product, we have \(a \cdot b = b \cdot a\).

Step 2: Analyzing Part (b)

In part (b), we have the expression \(a + (b + c)\). According to the Associative Property of Addition, which states that the way in which numbers are grouped in addition does not change the sum, we can rewrite it as \((a + b) + c\).

Step 3: Analyzing Part (c)

The expression in part (c) is \(a(b+c)\). By using the Distributive Property, which states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding, we have \(a(b+c) = ab + ac\).

Key Concepts

Commutative Property

The commutative property is a fundamental concept in mathematics, especially concerning real numbers. It essentially states that the order of numbers in addition or multiplication can be switched without affecting the sum or product. For example, if we consider the multiplication of two numbers:

• The expression \(a \cdot b = b \cdot a\) demonstrates the commutative property of multiplication.
• Similarly, for addition, \(a + b = b + a\).

In our original exercise, part (a) makes use of the commutative property of multiplication. It shows that even if you change the order of multiplying \(a\) and \(b\), the result remains the same. This property simplifies calculations and rearranges expressions without changing their meanings, making problem-solving more flexible.

Associative Property

The associative property applies to both addition and multiplication. It indicates that when three or more numbers are added or multiplied, the grouping of these numbers does not affect the result. Let's break this down:

• For addition, the associative property is shown as \((a + b) + c = a + (b + c)\).
• For multiplication, it manifests as \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).

In part (b) of the exercise, we used the associative property of addition. The expression \(a + (b + c)\) can be reorganized to \((a + b) + c\). Both expressions lead to the same sum. This property helps in simplifying complex arithmetic operations by allowing us to regroup numbers for easier computation.

Distributive Property

The distributive property bridges multiplication and addition, playing a critical role in algebra. It states that multiplying a number by a sum gives the same result as multiplying the number by each addend separately, then adding the products. In formula terms, we express it as:

• \(a(b + c) = ab + ac\).

In part (c) of our problem, notice how \(a(b + c)\) is resolved using this property. We distribute the \(a\) to both \(b\) and \(c\), resulting in the expression \(ab + ac\). This property is extremely useful for expanding algebraic expressions and simplifying calculations by breaking down complex problems into manageable parts.