Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the rational expression. \(\frac{\frac{6}{y}-\frac{4}{x}}{y}\)

Step-by-Step Solution

Answer

\(\frac{6x - 4y}{xy^2}\) is the simplified form.

Step 1: Identify the Rational Expression

The given expression is a complex fraction: \(\frac{\frac{6}{y}-\frac{4}{x}}{y}\). The goal is to simplify this to a single rational expression.

Step 2: Combine the Numerator

The numerator is \(\frac{6}{y} - \frac{4}{x}\). To combine these, find a common denominator. The common denominator of \(y\) and \(x\) is \(xy\). By rewriting, we get: \(\frac{6x}{xy} - \frac{4y}{xy} = \frac{6x - 4y}{xy}\).

Step 3: Simplify the Whole Expression

Now substitute the simplified numerator into the original expression: \(\frac{\frac{6x - 4y}{xy}}{y}\). This can be rewritten by multiplying the denominator \(y\): \(\frac{6x - 4y}{xy} \cdot \frac{1}{y} = \frac{6x - 4y}{xy^2}\).

Step 4: Final Simplification

The expression \(\frac{6x - 4y}{xy^2}\) does not have any common factors that can be canceled further, and thus is fully simplified.

Key Concepts

Complex Fractions

Complex fractions can be intimidating, but they simply involve fractions within fractions. In math, a complex fraction is defined as a fraction where either the numerator, the denominator, or both contain a fraction themselves. For example, in the expression \( \frac{\frac{6}{y}-\frac{4}{x}}{y} \), the part \( \frac{6}{y}-\frac{4}{x} \) is a complex numerator on its own. Dealing with complex fractions requires careful handling to simplify them into simpler expressions. This typically involves operations such as finding common denominators or even rewriting the expression entirely to eliminate the nested fractions. By understanding each component separately, you can unravel the complexity step by step.

Common Denominator

The idea of a common denominator might seem abstract at first, but it's really about finding a mutual basis for comparison. When we have two fractions, like \( \frac{6}{y} \) and \( \frac{4}{x} \), they have different denominators: \( y \) and \( x \), respectively. To add or subtract these fractions, we need to rewrite them so they share the same denominator, or in other words, find their Least Common Denominator (LCD). The LCD for \( y \) and \( x \) is \( xy \).

• Multiply \( \frac{6}{y} \) by \( \frac{x}{x} \) to get \( \frac{6x}{xy} \).
• Multiply \( \frac{4}{x} \) by \( \frac{y}{y} \) to get \( \frac{4y}{xy} \).

This process allows you to then subtract or add the numerators while focusing on a single, unified denominator.

Algebraic Fractions

Algebraic fractions extend the idea of fractions to include algebraic expressions in the numerator and/or denominator. In the exercise, fractions such as \( \frac{6}{y} \) and \( \frac{4}{x} \) are examples of algebraic fractions because they include variables. They behave the same way as their numeric counterparts in operations, but with the added nuance of dealing with variables.
Understanding algebraic fractions is important as they can be simplified further using similar techniques: finding common denominators, factoring expressions, and cancelling like terms if possible. The key is to remember that algebraic manipulation rules apply, such as distributing, combining like terms, or factorization to simplify any complex expressions.

Multiplying Fractions

Multiplying fractions might seem complicated, but it's one of the simpler operations to perform. For instance, taking the expression \( \frac{6x - 4y}{xy} \cdot \frac{1}{y} \), we multiply directly across both the numerators and denominators.
This involves:

• Multiplying the numerators: \((6x - 4y) \cdot 1 = 6x - 4y\).
• Multiplying the denominators: \(xy \cdot y = xy^2\).

This retains the rule of simplicity in multiplication over fractions: "Numerator with numerator, denominator with denominator." With practice, multiplying fractions, even algebraic ones, becomes intuitive, a core part of mastering rational expression simplification.