Problem 9 - Intermediate Algebra

Question

Complete each solution to simplify the rational expression. a. Fill in the blank: The expression \(\frac{\frac{a}{b}}{\frac{c}{d}}\) is equivalent to \(\frac{a}{b} \square \frac{c}{d}\) b. What is the numerator and what is the denominator of the following complex fraction? $$ \frac{6-k-\frac{5}{k}}{k^{2}-9} $$

Step-by-Step Solution

Answer

a. The blank is filled with \( \times \frac{d}{c} \). b. Numerator: \( 6-k-\frac{5}{k} \), Denominator: \( k^{2}-9 \).

Step 1: Understanding Complex Fractions

The given expression \( \frac{\frac{a}{b}}{\frac{c}{d}} \) is a complex fraction. Simplifying a complex fraction involves converting it into a simple fraction. This is done by dividing the numerator by the denominator, which is equivalent to multiplying the numerator by the reciprocal of the denominator.

Step 2: Applying the Concept to Part a

In part a, the problem asks us to fill in the blank of the expression \( \frac{\frac{a}{b}}{\frac{c}{d}} \) to find its equivalent form. Using the concept from Step 1, we rewrite the expression as \( \frac{a}{b} \times \frac{d}{c} \). Therefore, the blank should be filled with \( \times \frac{d}{c} \).

Step 3: Identifying Numerator and Denominator in Part b

The problem asks for the numerator and denominator of the complex fraction \( \frac{6-k-\frac{5}{k}}{k^{2}-9} \). First, simplify the numerator as much as possible: \( 6 - k - \frac{5}{k} \) is the numerator. The denominator is already in its simplest form: \( k^{2} - 9 \). Thus, the numerator is \( 6-k-\frac{5}{k} \) and the denominator is \( k^{2}-9 \).

Key Concepts

Complex Fractions

Complex fractions may sound advanced, but they actually break down into simpler steps. A complex fraction is a fraction where the numerator, the denominator, or both, are also fractions themselves. Typically, when you see a nested fraction like \( \frac{\frac{a}{b}}{\frac{c}{d}} \), it looks like a fraction inside a fraction. To simplify a complex fraction, you'll need to convert it into a single fraction.

This is accomplished by treating the main line of the fraction like a division sign. You are dividing the fraction \( \frac{a}{b} \) by the fraction \( \frac{c}{d} \). This division can be simplified by multiplying the numerator by the reciprocal of the denominator. This shifts the expression to: \( \frac{a}{b} \times \frac{d}{c} \). Remember, turning the division into multiplication by the reciprocal is a key step in simplifying complex fractions. It makes the expression more approachable and easy to work with.

Numerator and Denominator

Breaking down fractions involves understanding their basic parts—the numerator and the denominator. In any fraction \( \frac{m}{n} \), the numerator 'm' is the top number, and the denominator 'n' is the bottom number. These two components play specific roles:

• The numerator represents the number of parts we have.
• The denominator indicates the number of equal parts the whole is divided into.

When dealing with complex fractions, identifying these parts becomes crucial. For example, in the expression \( \frac{6-k-\frac{5}{k}}{k^{2}-9} \), the process remains the same. The entire expression \( 6-k-\frac{5}{k} \) is the numerator, which may include other fractions. While the entire expression \( k^{2}-9 \) is the denominator. Simplifying these parts individually, if possible, can make working with complex fractions easier.

Multiplying Reciprocals

A reciprocal is simply the 'flip' of a fraction. If you have a fraction like \( \frac{c}{d} \), its reciprocal is \( \frac{d}{c} \). Multiplying by a reciprocal is an essential tool when simplifying complex fractions. This method makes seemingly challenging expressions more manageable.

Why does this work? When you multiply a fraction by its reciprocal, the result is always 1. This is because the numerator and denominator effectively "cancel each other out". By changing the division of two fractions into the multiplication by the reciprocal, not only do you simplify the expression, but you also reveal a simpler relationship that can be easier to compute.

Whenever you encounter division with fractions, like in \( \frac{\frac{a}{b}}{\frac{c}{d}} \), remember this important step. Instead of dividing, multiply the numerator by the reciprocal of the denominator. This converts the complex fraction into a straightforward multiplication problem, leading to a simplified result.