Problem 47 - College Algebra with Corequisite Support

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

The numerator of the main fraction is \( \frac{a}{b} - \frac{b}{a} \). To simplify, we need a common denominator, which in this case would be \( ab \).The expression becomes:\[ \frac{a^2}{ab} - \frac{b^2}{ab} = \frac{a^2 - b^2}{ab} \].

Step 2: Factor the Difference of Squares

In the expression \( a^2 - b^2 \), we can apply the difference of squares identity, which states \( a^2 - b^2 = (a - b)(a + b) \).Thus, the fraction \( \frac{a^2 - b^2}{ab} \) can be rewritten as:\[ \frac{(a-b)(a+b)}{ab} \].

Step 3: Simplify the Denominator

The denominator of the main fraction is \( \frac{a+b}{ab} \). This cannot be simplified further, so we'll keep it as it is.

Step 4: Simplify the Division of Two Fractions

We now need to divide the simplified numerator by the simplified denominator:\[ \frac{\frac{(a-b)(a+b)}{ab}}{\frac{a+b}{ab}} \].When dividing by a fraction, multiply by its reciprocal:\[ \frac{(a-b)(a+b)}{ab} \times \frac{ab}{a+b} \].

Step 5: Cancel Common Terms

The terms \( ab \) in the numerator and denominator cancel out. Also, \( a + b \) is present in both the numerator and the denominator as well, allowing further simplification. So we are left with:\[ (a-b) \].

Step 6: Conclusion

Therefore, the simplified form of the original rational expression is \( a - b \).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic expression. It involves subtraction between two squared terms. For example, in the expression \(a^2 - b^2\), we see the squares of \(a\) and \(b\) being subtracted. This type of expression can be factored using the identity:

• \(a^2 - b^2 = (a - b)(a + b)\)

This identity is useful for simplifying many kinds of algebraic expressions. It shows that two squares can be broken down into a product of two binomials. In the context of rational expressions, recognizing a difference of squares can allow us to simplify more complex fractions by breaking them into simpler components.
Remembering this pattern is key. It not only simplifies expressions but also makes calculations more manageable.

Common Denominator

In fractions, especially in algebra, having a common denominator simplifies operations like addition or subtraction. Let's think about the example from your exercise. You start with \(\frac{a}{b} - \frac{b}{a}\). Before subtracting these fractions, they are transformed to have the same denominator:

• Identify the least common multiple (LCM) of the denominators, here \(ab\).
• Rewrite each fraction so that it has this common denominator.

So, \(\frac{a}{b}\) becomes \(\frac{a^2}{ab}\) and \(\frac{b}{a}\) turns into \(\frac{b^2}{ab}\). With the common denominator, both fractions can now be easily combined into one. Simplifying expressions this way is an essential skill in algebra, easing the process of manipulating and solving equations.

Fraction Division

Division of fractions often confuses learners because of its unique method. Instead of direct division, we multiply by the reciprocal. This concept appeared in your problem where the expression: \(\frac{\frac{(a-b)(a+b)}{ab}}{\frac{a+b}{ab}}\) had to be simplified:

• Change the division sign to multiplication and flip the second fraction.
• Therefore, divide by a fraction \(\frac{a+b}{ab}\) means multiplying by \(\frac{ab}{a+b}\).

This reciprocal swap changes a division problem into multiplication, making it easier to handle. Once flipped, treat the expression like any standard multiplication operation involving fractions. Simplifying becomes more straightforward as you cancel out common terms, such as \(ab\) and \(a+b\). Understanding this method is crucial when working with rational expressions, making complex operations simpler.

Algebraic Factorization

Algebraic factorization involves breaking down expressions into simpler, multiplied components. In your exercise, factorization simplified the expression \(a^2 - b^2\) into \((a-b)(a+b)\). Every factorization process follows certain rules and patterns:

• Identify common factors or special identities like the difference of squares.
• Use these patterns to rewrite expressions as products of other expressions.

For complex algebraic expressions, recognizing factorable patterns can aid in transforming and simplifying the work. It reduces large polynomial expressions into manageable pieces. Factorization not only makes expressions easier to handle but is also a critical process for solving algebraic equations. Once you master these patterns, simplifying rational expressions becomes much more intuitive.