Problem 44 - College Algebra

Question

For the following exercises, simplify the rational expression. $$ \frac{\frac{x}{4}-\frac{p}{8}}{p} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2x - p}{8p} \).

Step 1: Identify the Common Denominator

In the numerator, we have two fractions: \( \frac{x}{4} \) and \( \frac{p}{8} \). The common denominator for these fractions is 8 since it is the least common multiple of 4 and 8.

Step 2: Rewrite the Numerator with a Common Denominator

Rewrite \( \frac{x}{4} \) so that it has the common denominator of 8. This can be done by multiplying both the numerator and the denominator by 2: \( \frac{x}{4} = \frac{2x}{8} \). Now, you can combine the fractions: \( \frac{2x}{8} - \frac{p}{8} = \frac{2x - p}{8} \).

Step 3: Rewrite the Entire Expression

The rational expression now becomes: \( \frac{\frac{2x - p}{8}}{p} \).

Step 4: Simplify the Complex Fraction

Simplify the fraction by treating it as division: \( \frac{2x - p}{8} \times \frac{1}{p} = \frac{2x - p}{8p} \).

Step 5: Final Simplified Expression

The simplified rational expression is \( \frac{2x - p}{8p} \).

Key Concepts

Common Denominator

A common denominator is essential when you need to add or subtract fractions. It is a common multiple of the denominators of those fractions. This allows you to combine multiple fractions into a single fraction, making the simplification process smoother.
In the given exercise, the fractions \( \frac{x}{4} \) and \( \frac{p}{8} \) require a common denominator to be combined. The common denominator is crucial because it allows you to rewrite these fractions with equal denominators, enabling direct subtraction.

• For fractions like \( \frac{x}{4} \) and \( \frac{p}{8} \), find a number divisible by both denominators (4 and 8).
• Multiply both the numerator and the denominator of \( \frac{x}{4} \) by the same number to adjust it to this common denominator.

With a common denominator of 8, \( \frac{x}{4} \) becomes \( \frac{2x}{8} \), allowing straightforward subtraction from \( \frac{p}{8} \), leading to \( \frac{2x - p}{8} \). Using a common denominator simplifies rational expressions by creating uniformity.

Complex Fractions

Complex fractions have one or more fractions in their numerator, denominator, or both. They can be intimidating, but with the right approach, they are manageable. Simplifying a complex fraction often involves converting it into a simpler form. This involves treating the fraction as a division problem.
Consider the fraction in the exercise: \( \frac{\frac{2x - p}{8}}{p} \). This is a complex fraction, as the numerator itself is a fraction. To simplify, follow these steps:

• View the entire fraction as a division: It represents \( \frac{2x - p}{8} \) divided by \( p \).
• In division by a fraction, "flip" the second term and multiply: \( \frac{2x - p}{8} \times \frac{1}{p} \).
• The result is a simpler form: \( \frac{2x - p}{8p} \).

Simplifying complex fractions by turning division into multiplication (by the reciprocal of the denominator) is a powerful technique in algebra, simplifying fractions with multiple layers.

Least Common Multiple

The least common multiple (LCM) is essential for simplifying expressions involving multiple fractions. It is the smallest positive integer that is a multiple of each of the denominators you are working with.
Finding the LCM is particularly useful when you deal with fractions that have different denominators. In the exercise, you encountered two denominators: 4 and 8. Here's how to determine the LCM:

• List the multiples of each number until you reach a common value: Multiples of 4 are 4, 8, 12, 16, and so on, while multiples of 8 are 8, 16, 24, etc.
• The first common multiple they share is 8, making it the least common multiple.
• This allows both \( \frac{x}{4} \) and \( \frac{p}{8} \) to be adjusted to fractions that can be added or subtracted with ease.

Finding the least common multiple simplifies the process by ensuring that all fractions involved have the same denominator, simplifying further operations on the expression. This step is vital for creating a more unified expression that can be simplified effectively.