Problem 21 - Beginning and Intermediate Algebra

Question

Use the commutative and associative properties to simplify each expression. See Example 3 \(\frac{1}{5}(5 y)\)

Step-by-Step Solution

Answer

The expression simplifies to \(y\).

Step 1: Apply the Commutative Property

The commutative property states that you can change the order of factors in a multiplication without changing the product, so we express the term inside parenthesis differently:\[(5 \cdot y) = (y \cdot 5)\]

Step 2: Simplify the Inside Expression Using the Associative Property

The associative property allows us to regroup terms in a product. So we can remove the parenthesis in the original expression, using the transition already made: \[\frac{1}{5} \cdot (y \cdot 5) = (\frac{1}{5} \cdot y) \cdot 5 \]

Step 3: Simplify the Expression

Notice that multiplying \(\frac{1}{5}\) by 5 gives 1:\[(\frac{1}{5} \cdot 5)\] So the expression becomes:\[y \cdot 1 = y\]

Step 4: Final Step: Conclude the Simplification

Since multiplying by 1 does not change the value, we conclude the expression simplifies to:\[y\]

Key Concepts

Commutative Property

The commutative property is a fundamental principle in mathematics, particularly useful in algebraic simplifications. It states that the order of numbers in addition or multiplication does not affect the result. For multiplication, this means:

• If you swap the numbers, the answer remains the same: \(a \times b = b \times a\).
• This property can help rewrite expressions to better suit further simplification steps.

In our exercise, we applied the commutative property to change the order of multiplication inside the parenthesis \((5 \cdot y) = (y \cdot 5)\). By rearranging the terms, it sets the stage for the associative property to be applied more easily. This flexibility is incredibly powerful, making calculations more intuitive and often revealing simpler forms of expressions.

Associative Property

Associative property is another key property of addition and multiplication. It allows us to regroup numbers differently without changing the overall result. This property states that:

• For multiplication: \((a \times b) \times c = a \times (b \times c)\).
• It enables us to rethink and reorganize terms in an equation or expression.

In the example we examined, the associative property was strategically used after applying the commutative property. We regrouped the expression \(\frac{1}{5} \cdot (y \cdot 5)\) as \((\frac{1}{5} \cdot y) \cdot 5\). This clever adjustment set up our expression for the final simplification step, highlighting the importance of these properties in finding more workable forms.

Multiplication Simplification

Multiplication simplification often involves using fundamental properties like the commutative and associative properties to streamline expressions. Simplification aims to reduce complexity, making expressions easier to understand and calculate.

• In our problem, simplifying \(\frac{1}{5} \cdot 5\) to 1 is a crucial step.
• This application leads to a beautifully simple final outcome: \(y \cdot 1 = y\).

This result underscores another important simplification concept: any number multiplied by 1 remains unchanged. By organizing and simplifying terms preconceived as complex, like our given problem, we manage to convey that even elaborate expressions can be rewritten to appear elementary and direct.