Problem 7 - Elementary and Intermediate Algebra

Question

Find the product of the rational expression and its reciprocal. \(\frac{3}{x+2} \cdot \frac{x+2}{3}\)

Step-by-Step Solution

Answer

The product is 1.

Step 1: Write the Expression

Identify the given rational expression and its reciprocal. We have \(\frac{3}{x+2}\) and \(\frac{x+2}{3}\).

Step 2: Apply the Multiplication of Fractions

To multiply fractions, multiply the numerators together and the denominators together. This gives \(\frac{3}{x+2} \cdot \frac{x+2}{3} = \frac{3 \times (x+2)}{(x+2) \times 3}\).

Step 3: Simplify the Expression

Notice that both the numerator and the denominator are the same: \(3 \times (x+2)\). This means you can cancel out these identical terms, yielding \(\frac{3 \times (x+2)}{3 \times (x+2)} = 1\).

Key Concepts

Multiplication of Fractions

The process of multiplying fractions is a foundational skill in algebra. When you encounter a problem that involves multiplying fractions, such as rational expressions, the rule is straightforward:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

For instance, consider the fractions \( \frac{a}{b} \) and \( \frac{c}{d} \). To find the product, you multiply the numerators \( a \times c \) and the denominators \( b \times d \). The resulting fraction is \( \frac{a \times c}{b \times d} \).
This concept is the same when dealing with rational expressions such as \( \frac{3}{x+2} \) and its reciprocal \( \frac{x+2}{3} \). You apply the rule, giving \( \frac{3 \times (x+2)}{(x+2) \times 3} \). This multiplication step sets the stage for further simplification.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is crucial in algebra, especially when simplifying expressions or solving equations. The reciprocal of a fraction \( \frac{a}{b} \) is simply flipping the fraction, leading to \( \frac{b}{a} \). It's worth noting that a number multiplied by its reciprocal always equals 1.
This concept is very useful! Whenever you need to solve for a term that involves division, or you wish to simplify an equation, the reciprocal can simplify your work considerably. In the problem we're discussing, the reciprocal of \( \frac{3}{x+2} \) is \( \frac{x+2}{3} \).
By multiplying the fraction by its reciprocal, you get a beautifully simplified result of 1 because they effectively "cancel" each other out.

Simplifying Algebraic Expressions

In algebra, simplifying an expression means reducing it to its simplest form without changing its value. When you simplify expressions, you may perform operations like canceling common factors or combining like terms.
For rational expressions, simplification often involves identifying and canceling out common factors in the numerator and denominator. In this case, after multiplying \( \frac{3}{x+2} \) by its reciprocal \( \frac{x+2}{3} \), we arrive at \( \frac{3 \times (x+2)}{3 \times (x+2)} \).
Here, both the numerator and the denominator contain the same expression \( 3 \times (x+2) \). By canceling this out, the expression reduces to 1. This process shows how multiplication and reciprocal work together to simplify even complex expressions. By following these steps, you can manage rational expressions with confidence!