Problem 47 - College Algebra

Question

For the following exercises, 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

First, we'll simplify the numerator \(\frac{a}{b}-\frac{b}{a}\). To subtract these fractions, we find a common denominator, which is \(ab\). Thus, we rewrite the fractions as follows: \(\frac{a^2}{ab} - \frac{b^2}{ab} = \frac{a^2 - b^2}{ab}\).

Step 2: Rewrite the Denominator

Next, examine the denominator \(\frac{a+b}{ab}\). It is already a single fraction with denominator \(ab\).

Step 3: Write the Entire Expression as a Division Problem

Now express the original problem using the simplified numerator and denominator. We have: \[\frac{\frac{a^2 - b^2}{ab}}{\frac{a+b}{ab}}.\]

Step 4: Simplify by Dividing Fractions

To simplify, we divide the fractions by multiplying the numerator by the reciprocal of the denominator: \[ \frac{a^2 - b^2}{ab} \times \frac{ab}{a+b} = \frac{a^2 - b^2}{a+b}. \]

Step 5: Factor the Numerator

The expression \(a^2 - b^2\) is a difference of squares, which can be factored as \((a+b)(a-b)\). So, replace \(a^2 - b^2\) with \((a+b)(a-b)\) to get: \[\frac{(a+b)(a-b)}{a+b}.\]

Step 6: Cancel the Common Terms

In the fraction \(\frac{(a+b)(a-b)}{a+b}\), the \((a+b)\) terms in the numerator and denominator cancel each other out, simplifying to \(a-b\).

Step 7: Write the Final Simplified Expression

After canceling out the \((a+b)\) terms, the expression is simplified to \(a-b\).

Key Concepts

Common Denominator

When simplifying expressions with fractions, finding a common denominator is essential. The common denominator is the least common multiple of the denominators you're working with. In the context of rational expressions, this means that both fractions involved need to be expressed with the same base. This allows for straightforward subtraction or addition.

In the example we have, the expression \( \frac{a}{b} - \frac{b}{a} \) requires a common denominator. Since both denominators are \(a\) and \(b\), multiplying them gives us a common denominator of \(ab\). This allows us to rewrite the subtraction as:

• \( \frac{a^2}{ab} - \frac{b^2}{ab} \)

By converting them into equivalent fractions with the same denominator, we made it much easier to subtract and simplify further.

Difference of Squares

The difference of squares is a special algebraic expression that follows the form \(a^2 - b^2\). This can be factored into \((a+b)(a-b)\). This pattern is essential because it helps in simplifying expressions. Whenever you encounter a quadratic term subtracted by another, recognize it as a difference of squares.

In the example problem, we see that the numerator \(a^2 - b^2\) is a difference of squares. Recognizing this, we can factor it into:

• \((a+b)(a-b)\)

This step simplifies the expression and makes it possible to cancel out terms in the upcoming steps.

Factoring Algebraic Expressions

Factoring is the process of breaking down equations into simpler terms, which can be multiplied to give the original expression. It's an essential skill for simplifying complex expressions and solving equations efficiently.

In our rational expression example, we factored the difference of squares \(a^2 - b^2\) into \((a+b)(a-b)\). This step highlights the importance of recognizing patterns like the difference of squares to simplify an expression.

Upon factoring, we had the expression \(\frac{(a+b)(a-b)}{a+b}\). This made it possible to cancel the common factor of \((a+b)\) present in both the numerator and the denominator, leaving us with \(a-b\). By factoring expressions effectively, simplification becomes not only easier but also clearer.