Problem 30 - Introductory Algebra for College Students

Question

Simplify complex rational expression by the method of your choice. \(\frac{\frac{8}{x^{2}}-\frac{2}{x}}{\frac{10}{x}-\frac{6}{x^{2}}}\)

Step-by-Step Solution

Answer

\(\frac{4 - x}{5x - 3}\)

Step 1: Find a Common Denominator for Each Fraction

The first step here is to find a common denominator for each fraction. Start with the numerator of the complex fraction: \(\frac{8}{x^{2}}-\frac{2}{x}\). The common denominator here is \(x^{2}\). As for the denominator of the complex fraction: \(\frac{10}{x}-\frac{6}{x^{2}}\). The common denominator is also \(x^{2}\).

Step 2: Rewrite The Fractions to Have the Same Denominator

Rewrite these fractions to have the same denominator. For the numerator: \(\frac{8}{x^{2}}-\frac{2x}{x^{2}}\) simplifies to \(\frac{8 - 2x}{x^{2}}\). For the denominator: \(\frac{10x}{x^{2}}-\frac{6}{x^{2}}\) simplifies to \(\frac{10x-6}{x^{2}}\)

Step 3: Apply Division Rule

We know that \(\frac{a}{c} ÷ \frac{b}{c} = \frac{a}{b}\) by the division rule of fractions. Applying the same rule, our expression becomes \((\frac{8 - 2x}{x^{2}}) ÷ (\frac{10x - 6}{x^{2}})\), which simplifies to \(\frac{8 - 2x}{10x - 6}\)

Step 4: Simplify Further

Looking at the simplified fraction, both the numerator and the denominator have common factors. By dividing both by 2 we get: \(\frac{4 - x}{5x - 3}\). Which is our final answer.

Key Concepts

Common Denominator

When working with fractions, a common denominator is essential to add or subtract the fractions seamlessly. A common denominator is the same as having a common 'floor' upon which differences in the 'heights' of numerators can be weighed appropriately. In simpler terms, the common denominator allows fractions to be expressed in terms of the same size share.

Here's how you can determine the common denominator for each fraction:

• Identify the denominators in each fraction. In our exercise, the fractions found in both the numerator and denominator of the complex expression have denominators of either \(x\) or \(x^2\).
• Find the least common denominator that can accommodate all individual denominators. Here, the common denominator for both parts is \(x^2\).

After determining this, you can rewrite each fraction with this common denominator, allowing for easy subtraction or addition of numerators. This step is crucial because it sets the stage for simplifying the complexities of the expression later on.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator no longer have any common factors other than 1. By doing this, we make the fraction easier to understand and use.

In simplifying a body of fractions in our problem:

• Substitute the original fractions by rewriting them with the common denominator. For instance, from our exercise: \( \frac{8}{x^2} - \frac{2x}{x^2} \).
• Simplify the numerators by performing basic arithmetic operations. The expression becomes \( \frac{8 - 2x}{x^2} \).
• Repeat similarly for the denominator of the entire complex fraction, leading to \( \frac{10x - 6}{x^2} \).

The essence here is reducing the fractions without losing the value they're supposed to express. This systematic approach of simplifying common denominators helps unravel complex fractions.

Division Rule of Fractions

The division of fractions is a method where one fraction divides another, using a specific rule. The rule is: when you divide by a fraction, it is equivalent to multiplying by its reciprocal. This can make things much clearer in mathematics, especially with complex expressions.

In our complex fraction, the exercise proceeds to utilize this rule:

• Rewrite the expression from a division of fractions into a singular fraction. Use the division rule: \( \frac{a}{c} \div \frac{b}{c} = \frac{a}{b} \).
• By applying this rule to \( \frac{8 - 2x}{x^2} \div \frac{10x - 6}{x^2} \), we eliminate the common denominator (\(x^2\)), simplifying it to \( \frac{8 - 2x}{10x - 6} \).

This step is very significant because it transforms a two-fraction division into a single simpler fraction, easing further simplification steps. The division rule of fractions serves as a wonderful trick to cut down complex steps in fraction-based problems.