Problem 22 - Algebra A Combined Function

Question

Use the commutative and associative properties to simplify each expression. See Examples 5 and 6. $$ \frac{1}{8}(8 z) $$

Step-by-Step Solution

Answer

The expression simplifies to \( z \).

Step 1: Recall the Commutative Property

The commutative property of multiplication states that the order in which two numbers are multiplied does not change the product. This means that \( a \times b = b \times a \). In this problem, we will apply this property to the multiplication inside the expression \( \frac{1}{8}(8z) \).

Step 2: Recall the Associative Property

The associative property of multiplication states that how you group numbers when multiplying does not change the product. This means that \( (a \times b) \times c = a \times (b \times c) \). In this problem, since there are only two factors, grouping isn't necessary, but understanding this principle can help simplify complex expressions in later steps.

Step 3: Simplifying the Expression

Apply the commutative property to rewrite \( \frac{1}{8}(8z) \) as \( (\frac{1}{8} \times 8) \times z \).

Step 4: Simplify the Multiplication

Evaluate \( \frac{1}{8} \times 8 \). By definition, \( \frac{1}{8} \times 8 = 1 \) because any number multiplied by its reciprocal equals one.

Step 5: Final Simplified Expression

Now that \( \frac{1}{8} \times 8 = 1 \), substitute back into the expression to get \( 1 \times z \), which simply equals \( z \). Therefore, the expression \( \frac{1}{8}(8z) \) simplifies to \( z \).

Key Concepts

Commutative Property

The commutative property is a fundamental principle in algebra that is particularly useful for simplifying expressions. It states that the order of numbers in a multiplication operation can be changed without affecting the result. This means if you have two numbers, say \( a \) and \( b \), it doesn't matter whether you multiply them as \( a \times b \) or \( b \times a \)—the product will remain the same.

• This property makes it easy to rearrange numbers to simplify calculations.
• It's helpful when trying to see certain factors or pair numbers that might be easier to multiply together first.

In the case of the expression \( \frac{1}{8}(8z) \), the commutative property allows us to rearrange the terms to \( (\frac{1}{8} \times 8) \times z \). This strategically aligns the reciprocal factors \( \frac{1}{8} \) and \( 8 \), making the simplification straightforward.

Associative Property

The associative property of multiplication tells us that when multiplying multiple numbers, the way in which the numbers are grouped doesn't change the outcome. It can be looked at as the property where parentheses can be moved around in an expression without altering the result. For multiplication, this means: \( (a \times b) \times c = a \times (b \times c) \).

• The associative property helps in simplifying and understanding complex expressions.
• It ensures consistency when dealing with chains of multiplication involving more than two numbers.

Though our specific example, \( \frac{1}{8}(8z) \), doesn't showcase grouping because it involves only two factors, the background understanding of associative property is crucial. It suggests that when more numbers are involved, we can re-group them for easier computations without altering the result. This knowledge is foundational for more complex algebraic manipulations.

Multiplication of Fractions

The multiplication of fractions might seem slightly intricate at first, but it follows straightforward rules. To multiply fractions, you multiply the numerators together and the denominators together. For example, multiplying \( \frac{a}{b} \) by \( \frac{c}{d} \) results in \( \frac{a \times c}{b \times d} \).
In our task of simplifying \( \frac{1}{8}(8z) \), understanding fractions becomes more intuitive with the recognition of reciprocals. A reciprocal of a number is what you multiply that number by to get 1. For instance, the reciprocal of \( 8 \) is \( \frac{1}{8} \), and vice versa.

• By multiplying a number by its reciprocal, such as \( \frac{1}{8} \times 8 \), you end up with 1.
• This simplifying step leads to a much simpler expression overall, as in our case: it reduces \( (\frac{1}{8} \times 8) \times z = 1 \times z \).

These rules make working with fractions manageable and are integral to algebraic simplification processes.