Problem 2 - Precalculus Mathematics for Calculus

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) \(a b = b a\); Commutative Property of Multiplication. (b) \(a + (b + c) = (a + b) + c\); Associative Property of Addition. (c) \(a(b+c) = ab + ac\); Distributive Property.

Step 1: Understanding the problem

We need to complete each statement using the correct expression and identify the property of real numbers that justifies our completion. This requires knowing basic properties of arithmetic and real numbers.

Step 2: Complete statement (a)

For statement (a) \( a b = \), the property involved is the Commutative Property of Multiplication. According to this property, changing the order of factors does not change the product. Therefore, \( a b = b a \).

Step 3: Name the property for statement (a)

The property used in Step 2 is the Commutative Property of Multiplication. The final expression for (a) is \( a b = b a \); Commutative Property of Multiplication.

Step 4: Complete statement (b)

For statement (b) \( a + (b + c) = \), the property involved is the Associative Property of Addition. According to this property, the way in which numbers are grouped does not change their sum. Therefore, \( a + (b + c) = (a + b) + c \).

Step 5: Name the property for statement (b)

The property used in Step 4 is the Associative Property of Addition. The final expression for (b) is \( a + (b + c) = (a + b) + c \); Associative Property of Addition.

Step 6: Complete statement (c)

For statement (c) \( a(b+c) = \), the property involved is the Distributive Property. According to this property, multiplying a number by a sum is the same as doing each multiplication separately. Therefore, \( a(b + c) = ab + ac \).

Step 7: Name the property for statement (c)

The property used in Step 6 is the Distributive Property. The final expression for (c) is \( a(b+c) = ab + ac \); Distributive Property.

Key Concepts

Commutative Property

The Commutative Property is a fundamental concept in mathematics that applies to both addition and multiplication.

• Addition: The order of numbers does not change the result. For example, \( a + b = b + a \). Both expressions result in the same sum. This might seem simple, but it is a crucial idea that allows us to rearrange terms for simplification and problem-solving.
• Multiplication: Similarly, for multiplication, \( a \times b = b \times a \). This means that rearranging the factors does not affect the product. This property is widely used in algebra to simplify expressions and solve equations.

Imagine you have 3 apples and 2 oranges. Whether you count the apples first or the oranges first, you still end up with the same number of fruits. The Commutative Property helps us understand that order in addition and multiplication is not important when it comes to determining the total or the product.

Associative Property

The Associative Property deals with grouping numbers when adding or multiplying them. It's all about how numbers are associated or grouped.

• Addition: Consider this property as the rule that lets you move parentheses around. For example, \( (a + b) + c = a + (b + c) \). The sum remains unchanged regardless of how the numbers are grouped.
• Multiplication: Similarly, for multiplication, \( (a \times b) \times c = a \times (b \times c) \). The product does not change with different groupings of the factors.

An everyday analogy would be organizing tasks. If you have three chores to complete, you might group them differently on different days, but the total work completed remains the same. The Associative Property reassures us that how we group numbers does not affect the outcome of addition or multiplication.

Distributive Property

The Distributive Property connects addition and multiplication. It allows us to distribute a multiplying factor across a sum inside parentheses.

• When you see an expression like \( a(b + c) \), the Distributive Property tells you that you can "distribute" \( a \) across the terms inside the parentheses: \( a \times b + a \times c \).
• This property is especially useful in algebra for expanding expressions and simplifying equations.

To visualize the Distributive Property, think of assembling identical kits for two groups. If each kit contains 3 pencils and 2 erasers, then distributing is like multiplying the number of kits by the number of pencils and erasers separately before adding them together. So, building 4 kits results in \( 4 \times (3 + 2) = 4 \times 3 + 4 \times 2 \) items in total. It's an efficient way to break down complex problems and make calculations simpler.