Problem 32 - Beginning and Intermediate Algebra

Question

Decide whether each statement is an example of a commutative, an associative, an identity, \(a n\) inverse, or the distributive property. $$ \frac{5}{8}\left(\frac{8}{5}\right)=1 $$

Step-by-Step Solution

Answer

Inverse property

Step 1: Understand the Given Statement

Look at the expression \(\frac{5}{8}\frac{8}{5}\rightarrow1\). It involves multiplication.

Step 2: Consider the Properties of Operations

There are several properties to consider: commutative, associative, identity, inverse, and distributive. Identify which of these properties involves multiplication or division.

Step 3: Recognize the Use of Inverse Property

The expression \(\frac{5}{8}\frac{8}{5}\rightarrow 1\) simplifies to 1 because \(\frac{5}{8}\) and \( \frac{8}{5}\) are multiplicative inverses. The inverse property states that for any non-zero number \(a\), we have \(\frac{a}{b} \times \frac{b}{a} = 1\)

Key Concepts

commutative property

The commutative property states that the order in which we add or multiply numbers does not change the result. In mathematical terms, it means:
For addition: \(a + b = b + a\)
For multiplication: \(a \times b = b \times a\)
For example:

• Added (4 + 2) = (2 + 4)
• Multiplied (3 \times 5) = (5 \times 3)

As you can see, swapping the numbers around doesn't change their sum or product. This property is very handy when solving algebraic equations as it allows flexibility in rearranging terms.

associative property

The associative property describes how the grouping of numbers does not affect their sum or product. In other words, how you place the parentheses does not change the result:
For addition: \((a + b) + c = a + (b + c)\)
For multiplication: \((a \times b) \times c = a \times (b \times c)\)
Examples include:

• Added: \((4 + 3) + 2 = 4 + (3 + 2)\)
• Multiplied: \((2 \times 3) \times 5 = 2 \times (3 \times 5)\)

Notice that no matter how the numbers are grouped, the sum or product remains the same. This property helps in simplifying complex expressions and solving equations.

identity property

The identity property involves the identity elements for addition and multiplication. These identity elements are the numbers that do not change the value of other numbers when used in an operation:
For addition, the identity element is 0: \(a + 0 = a\)
For multiplication, the identity element is 1: \(a \times 1 = a\)
Examples include:

• Addition: \(7 + 0 = 7\)
• Multiplication: \(9 \times 1 = 9\)

The identity property is important because it helps confirm that the operation does not alter the original number.

inverse property in algebra

The inverse property in algebra refers to the concept of adding or multiplying by an inverse to get the identity element.
For addition, the inverse of a number \(a\) is \(-a\). Together, they sum to zero: \(a + (-a) = 0\)

For multiplication, the inverse of a non-zero number \(a\) is \(\frac{1}{a}\). When multiplied, they result in one: \(a \times \frac{1}{a} = 1\)

For example, in multiplication:

• \(\frac{5}{8} \times \frac{8}{5} = 1\)

Notice that when the numerator and denominator are flipped, and multiplied, they equal 1.

distributive property

The distributive property connects addition and multiplication. It states that multiplying a number by the sum of two other numbers is the same as doing each multiplication separately and then adding the results. Mathematically:
\(a \times (b + c) = (a \times b) + (a \times c)\)
Another form:
\((a + b) \times c = (a \times c) + (b \times c)\)
Examples include:

• \(3 \times (4 + 5) = (3 \times 4) + (3 \times 5)\)
• \((7 + 2) \times 6 = (7 \times 6) + (2 \times 6)\)

The distributive property is often used to simplify algebraic expressions and solve equations.