Problem 11 - Intermediate Algebra

Question

Complete each property of multiplication. Then give its name. a. \(a \cdot b=b \cdot\square\) b. \((a b) c=\square\) c. \(0 \cdot a=\square\) d. \(1 \cdot a=\square\) e. \(a\left(\frac{1}{a}\right)=\square\)

Step-by-Step Solution

Answer

a. Commutative Property, b. Associative Property, c. Zero Property, d. Identity Property, e. Inverse Property.

Step 1: Completing property a

For the property \(a \cdot b = b \cdot \square\), recognize that this represents the Commutative Property of Multiplication, where the order of factors does not change the product. Therefore, \(a \cdot b = b \cdot a\). The missing term is \(a\).

Step 2: Completing property b

For \((ab) c = \square\), recognize this as the Associative Property of Multiplication, which allows the grouping of factors to change without affecting the product. Thus, it becomes \((ab) c = a(bc)\). The missing term is \(a(bc)\).

Step 3: Completing property c

The equation \(0 \cdot a = \square\) follows the Zero Property of Multiplication, where any number multiplied by zero results in zero. Thus, \(0 \cdot a = 0\). The missing term is \(0\).

Step 4: Completing property d

The equation \(1 \cdot a = \square\) represents the Identity Property of Multiplication, where any number multiplied by one remains unchanged. Hence, \(1 \cdot a = a\). The missing term is \(a\).

Step 5: Completing property e

For the property \(a\left(\frac{1}{a}\right) = \square\), this is the Multiplicative Inverse Property. A number multiplied by its reciprocal equals one. Therefore, \(a\left(\frac{1}{a}\right) = 1\). The missing term is \(1\).

Key Concepts

Commutative Property

The commutative property of multiplication states that changing the order of the numbers you are multiplying does not change the product. For two numbers, say \(a\) and \(b\), it implies that \(a \cdot b = b \cdot a\).
This is easy to see with numbers: if you take 2 times 3, you get 6. Similarly, if you do 3 times 2, you still get 6. This property can be a handy memory tool when solving problems, as it means the factors can be swapped without affecting the result.

Associative Property

The associative property of multiplication focuses on how numbers are grouped in a multiplication problem. Essentially, it states that no matter how you group the factors, the product will be the same. For example, for three numbers \(a\), \(b\), and \(c\), it means that

• \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)

This property is particularly useful when you're trying to simplify complex multiplication.
If you have numbers like 2, 3, and 4, you can easily verify this property: Calculate \((2 \cdot 3) \cdot 4\) and then \(2 \cdot (3 \cdot 4)\). In both cases, the result is 24.

Zero Property

The zero property of multiplication is straightforward but crucial: any number multiplied by zero is always zero. This property can be expressed as \(a \cdot 0 = 0\) or \(0 \cdot a = 0\).
Think of this as a practical tool. No matter how big the number is, if you multiply it by zero, you are left with zero. This concept is useful when simplifying expressions or equations because it allows us to quickly identify terms that contribute nothing to the product.

Identity Property

The identity property of multiplication states that any number multiplied by one remains unchanged. Symbolically, we write this as \(a \cdot 1 = a\) or \(1 \cdot a = a\).

• Consider 7 as \(a\). If you multiply 7 by 1, the result is still 7.

This property is helpful when we simplify equations because it allows us to remove unnecessary multiplicative factors of one. This helps keep our work neat and manageable.

Multiplicative Inverse

The multiplicative inverse, or reciprocal, is a fundamental concept for division. For any non-zero number \(a\), its multiplicative inverse is \(\frac{1}{a}\). When a number is multiplied by its inverse, the product is always one: \(a \cdot \frac{1}{a} = 1\).
For example, the multiplicative inverse of 5 is \(\frac{1}{5}\). Multiplying 5 by \(\frac{1}{5}\) results in 1. This property is used frequently in solving equations, especially when dividing by a fraction, since it helps transform a division into a multiplication problem for easier computation.
Understanding these inverse relationships is key to mastering fractions and can make seemingly complex problems much simpler.