Problem 27 - Algebra A Combined Function

Question

Use the commutative and associative properties to simplify each expression. See Examples 5 and 6. $$ \frac{3}{4}\left(\frac{4}{3} s\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( s \).

Step 1: Identify Properties

The commutative property allows numbers to be multiplied in any order. The associative property allows grouping of numbers in a multiplication operation to be changed. In this expression, both properties can be used to simplify the multiplication: \( \frac{3}{4}\left(\frac{4}{3} s\right) \).

Step 2: Rearrange Using Commutative Property

Using the commutative property of multiplication, rearrange the expression as follows: \( \left(\frac{3}{4} \times \frac{4}{3}\right) \times s \). This positions the fractions \( \frac{3}{4} \) and \( \frac{4}{3} \) next to each other.

Step 3: Simplify Using Associative Property

Apply the associative property to focus on simplifying the fractional multiplication: \( \frac{3}{4} \times \frac{4}{3} = \frac{3 \times 4}{4 \times 3} = \frac{12}{12} = 1 \).

Step 4: Simplify the Expression

Now, place the simplified product (1) in the expression with \( s \): \( 1 \times s = s \). Therefore, the simplified expression is \( s \).

Key Concepts

Associative Property

The associative property is a fundamental concept in mathematics. It deals with how numbers are grouped in operations. When you're multiplying numbers, the way in which you group them doesn't affect the final result. This is the associative property of multiplication.
Think of it like this: if you have three numbers, such as \( a, b, \) and \( c \), you can multiply them as \((a \times b) \times c\) or as \(a \times (b \times c)\). Either way, the product is the same.

• This means you can choose to group numbers in a way that makes calculations easier.
• Grouping often helps to simplify complex expressions and allows easier mental calculations.

In the context of fraction multiplication from the exercise, the associative property permits us to tackle fractions first, making the expression manageable. Rearranging parts of an expression so that the fractions are multiplied separately can lead to simplifications that remove the fractional component entirely.

Fraction Multiplication

Multiplying fractions can seem daunting at first, but it's actually quite straightforward. The key is to multiply the numerators together and the denominators together. In a fraction \( \frac{a}{b} \times \frac{c}{d} \), the multiplication would be \( \frac{a \times c}{b \times d} \).
An important tool in fraction multiplication is simplification. This involves canceling out any common factors between the numerator and the denominator before multiplying.

• This can reduce the numbers you're working with early, making calculations simpler.
• Always try to simplify wherever possible, especially when dealing with large numbers.

In our example, \( \frac{3}{4} \times \frac{4}{3} \) simplifies directly to \(1\) because the numerators and denominators are cross-multiplicative inverses, which means they cancel each other out, leaving \(1\). This simplification is key to eliminating complex parts of mathematical expressions.

Mathematical Expressions

Understanding mathematical expressions is crucial for solving problems. They consist of numbers, variables, and operators (like addition or multiplication) that work together to show a relationship. The exercise demonstrates how properties like commutative and associative can work to simplify these expressions.
Each component of an expression plays a role:

• **Numbers**: These can be constants or coefficients in expressions.
• **Variables**: Symbols, often letters, that represent numbers. They can vary or remain fixed based on context.
• **Operators**: Signs like +, -, ×, and ÷ that denote the mathematical operation to perform.

To simplify mathematical expressions, it's essential to recognize and correctly apply mathematical properties, such as those seen in the original exercise. Doing so not only provides clarity but also ensures calculations are as straightforward as possible. In the exercise, proper use of the associative and commutative properties leads to a clean and simple result, showing how effective simplification can be.