Problem 70 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{y}{x}+x^{-1}}{x^{-1}+\frac{2 x}{y}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{y(y+1)}{y + 2x^2} \).

Step 1: Simplifying the Numerator

The numerator of the complex fraction is \( \frac{y}{x} + x^{-1} \). We can rewrite \( x^{-1} \) as \( \frac{1}{x} \). So the numerator becomes \( \frac{y + 1}{x} \).

Step 2: Simplifying the Denominator

The denominator of the complex fraction is \( x^{-1} + \frac{2x}{y} \). We can rewrite \( x^{-1} \) as \( \frac{1}{x} \). Therefore, the denominator becomes \( \frac{1}{x} + \frac{2x}{y} \).

Step 3: Finding Common Denominators

For the numerator, \( \frac{y+1}{x} \) is already simplified. For the denominator \( \frac{1}{x} + \frac{2x}{y} \), we find a common denominator which is \( xy \). So: \[ \frac{1}{x} = \frac{y}{xy}, \quad \frac{2x}{y} = \frac{2x^2}{xy} \] So the denominator becomes \( \frac{y + 2x^2}{xy} \).

Step 4: Simplifying the Overall Fraction

Now we simplify the overall fraction: \[ \frac{\frac{y+1}{x}}{\frac{y + 2x^2}{xy}} \] by multiplying by the reciprocal of the denominator: \[ = \frac{y+1}{x} \cdot \frac{xy}{y + 2x^2} \]

Step 5: Canceling Common Factors

In the expression \( \frac{y+1}{x} \cdot \frac{xy}{y + 2x^2} \), the \( x \) in the numerator and \( x \) in the denominator cancel each other out, leaving: \[ \frac{y(y+1)}{y + 2x^2} \]

Step 6: Final Simplification and Result

The expression \( \frac{y(y+1)}{y + 2x^2} \) is fully simplified as there are no further common factors to cancel.

Key Concepts

Rational Expressions

Rational expressions are like fractions, but instead of whole numbers, they have polynomials in the numerator and/or the denominator. Understanding rational expressions is key when dealing with complex fractions because they allow us to manipulate expressions systematically. Complex fractions are fractions where both the numerator and the denominator are themselves fractions. So, they are like a fraction within a fraction!
When you see a complex fraction, the first step is usually to make the smaller fractions' denominators alike. This allows you to transform them into a single fraction, making it easier to manage.

• Identify the smaller fractions and rewrite them with a common denominator.
• Combine the fractions in the numerator and denominator separately using the least common denominator (LCD).

Grasping this technique will open up rational expressions for more detailed algebraic analysis, helping you solve more complex problems with ease.

Simplification

Simplifying complex fractions involves a step-by-step process that reduces the expression to its most straightforward form. Here's how:
To simplify a complex fraction like the one given, we first rewrite terms, such as converting negative exponents like \( x^{-1} \) into fractions, \( \frac{1}{x} \). This converts all terms into a consistent form and makes the fractions easier to manage.
Simplification often involves finding common denominators, especially when the terms involve division themselves. Let's make it simple:

• Once fractions in the numerator and denominator are combined individually, identify a common denominator for each.
• This helps in expressing each part as a single fraction, setting up the problem for further simplification.
• The final step: transform the complex fraction into a simple fraction by multiplying by the reciprocal of the denominator.

Through simplification, you're peeling off layers of complexity and revealing the true form of the expression. It's all about breaking things down into simpler, more manageable pieces.

Inverse Operations

Inverse operations are like the mathematical equivalent of undoing an action. They are crucial in algebra, as they help reverse calculations to get back to the original values. In the case of fractions, the inverse operation is really about reciprocals. This is especially relevant in dividing fractions—think 'flipping and multiplying.'

To simplify a complex fraction, we use the inverse operation of division by converting it into multiplication:

• For instance, dividing by \( \frac{y + 2x^2}{xy} \) becomes multiplying by \( \frac{xy}{y + 2x^2} \).
• Applying this operation allows us to simplify expressions without changing their value.

This understanding of inverse operations and how to use them in the context of fractions is invaluable. It allows students to attack complex algebraic problems with confidence and accuracy by turning them into simpler multiplication problems.