Problem 47 - College Algebra

Question

Simplify the rational expression. $$ \frac{\frac{a}{b}-\frac{b}{a}}{\frac{a+b}{a b}} $$

Step-by-Step Solution

Answer

The simplified expression is \( a-b \).

Step 1: Simplify the Numerator

The numerator of the given rational expression is \( \frac{a}{b} - \frac{b}{a} \). To simplify this, find a common denominator, which is \( ab \). Rewrite the expression as \( \frac{a^2}{ab} - \frac{b^2}{ab} \). This simplifies to \( \frac{a^2 - b^2}{ab} \).

Step 2: Simplify the Denominator

The denominator of the given rational expression is \( \frac{a+b}{ab} \). This is already in a simple form as there are no further simplifications possible at this step.

Step 3: Rewrite the Full Expression

Substitute the simplified forms of the numerator and the denominator back into the original expression to get \( \frac{\frac{a^2-b^2}{ab}}{\frac{a+b}{ab}} \).

Step 4: Divide the Fractions

To divide two fractions \( \frac{x}{y} \) and \( \frac{z}{w} \), multiply the first fraction by the reciprocal of the second. Thus, \( \frac{\frac{a^2-b^2}{ab}}{\frac{a+b}{ab}} = \frac{a^2-b^2}{ab} \times \frac{ab}{a+b} \).

Step 5: Cancel Common Factors

Cancel the common factor \( ab \) from the numerator and the denominator, resulting in \( \frac{a^2 - b^2}{a + b} \).

Step 6: Recognize Difference of Squares

Notice that \( a^2 - b^2 \) is a difference of squares and can be factored into \( (a-b)(a+b) \). Replace \( a^2 - b^2 \) with \( (a-b)(a+b) \) and express it as \( \frac{(a-b)(a+b)}{a+b} \).

Step 7: Final Simplification

Cancel out the common factor \( a+b \) from the numerator and denominator, leaving \( a-b \) as the simplified form of the expression.

Key Concepts

Simplifying Fractions

Simplifying fractions, whether they are simple numerical fractions or more complex algebraic expressions, is an essential skill in dealing with rational expressions. The basic idea is to reduce the fraction to its simplest form by canceling out common factors in the numerator and the denominator.

In the given exercise, we start with a complex fraction that involves multiple expressions in both the numerator and the denominator. The key steps to simplify include:

• Finding a common denominator for terms in the numerator or denominator.
• Rewriting the expressions using this common denominator.
• Canceling out common factors that appear in both the numerator and the denominator.

By following these steps, we ensure that the fraction is in its simplest possible form, making it easier to work with and understand.

Difference of Squares

A difference of squares is a special algebraic identity used frequently in mathematics, especially in simplifying and factoring expressions. It takes the form \( a^2 - b^2 \), and it can be factored as \( (a-b)(a+b) \).

The exercise involves recognizing this pattern within the expression \( \frac{a^2 - b^2}{a + b} \) during the simplification process. Here's how it works:

• Identify if the expression fits the \( a^2 - b^2 \) model.
• Factorize it into \( (a-b)(a+b) \), which allows us to cancel out \( a+b \) if it's also present in the denominator.
• This simplifies the expression significantly, reducing complexity.

Understanding differences of squares helps streamline the manipulation and simplification process of rational expressions and many other algebraic operations.

Factoring

Factoring is a method of rewriting an expression as a product of its simpler components, which is immensely useful for simplifying rational expressions. It involves breaking down complex expressions into multiplicative building blocks, which can often reveal common factors that can be canceled.

In our problem, factoring \( a^2 - b^2 \) into \( (a-b)(a+b) \) is a direct example of this process. Steps to factor an expression typically include:

• Looking for common factors across all terms.
• Recognizing special factoring patterns such as the difference of squares, perfect square trinomials, or others.
• Rewriting the expression as a product of these factors.

Factoring greatly facilitates the simplification of expressions by laying bare the underlying structure, making it easier to identify and cancel out terms, as we saw in the final simplification to \( a-b \) of the original expression.