Problem 34 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{\frac{y}{x+y}-\frac{x}{x-y}}{\frac{x}{x+y}+\frac{y}{x-y}} $$

Step-by-Step Solution

Answer

Question: Simplify the following complex rational expression: $$ \frac{\frac{y}{x+y} - \frac{x}{x-y}} {\frac{x}{x-y} + \frac{y}{x+y}} $$ Answer: $$ \frac{x(x - y)}{x^2 - xy} $$

Step 1: Find a common denominator

For both the numerator and denominator, the common denominator will be the product of the two denominators \((x+y)(x-y)\).

Step 2: Combine the numerators and denominators using the common denominator

Rewrite each fraction in the expression using the common denominator and then combine them: $$ \frac{\frac{y(x-y)}{(x+y)(x-y)} - \frac{x(x+y)}{(x+y)(x-y)}} {\frac{x(x-y)}{(x+y)(x-y)} + \frac{y(x+y)}{(x+y)(x-y)}} $$

Step 3: Simplify the expression

Simplify the numerator and denominator: $$ \frac{yx^2 - xy^2 - x^2 y - x^3} {x^3 - x^2 y + xy^2 - y^3} $$

Step 4: Look for any further simplification or factors that may cancel out

Notice that in the numerator and the denominator we can factor out \(x\) and that they share the factor \(x^2 - xy\): $$ \frac{x(x - y)(x + y)}{(x + y)(x^2 - xy)} $$ Now, we can cancel out the common factor of \((x+y)\) from the numerator and denominator: $$ \frac{x(x - y)}{x^2 - xy} $$ The final simplified complex rational expression is: $$ \frac{x(x - y)}{x^2 - xy} $$

Key Concepts

Simplification of Expressions

When faced with a complex rational expression, the goal is to simplify it to a more manageable form. Simplification helps to reveal any hidden patterns or solutions that might not be evident at first glance. To start, look for any opportunities to combine like terms or cancel out factors.

In the given exercise, you have a complex rational expression in the form of a large fraction, where both the numerator and the denominator are fractions themselves. The key to simplification here is to turn the complex expression into a simple one by reducing it. This often involves eliminating fractions by finding a common denominator, which we'll explore more in the next section.

• Identify the main components of the expression.
• Break down these components to their simplest forms by combining like terms or reducing fractions.
• Watch for any possibilities where terms may cancel each other out.

Always keep an eye out for potential simplifications at each step, as this can help in reaching the solution faster.

Common Denominator

Understanding common denominators is crucial when simplifying complex fractions. A common denominator is a shared multiple of the denominators within an expression.

In our complex rational expression, the fraction in both the numerator and the denominator has a different base. To simplify these, the first step is to find a common denominator, which will allow you to combine the fractions. For our expression \[\frac{\frac{y}{x+y}-\frac{x}{x-y}}{\frac{x}{x+y}+\frac{y}{x-y}},\]we identified \((x+y)(x-y)\)as the common denominator.

• This allows you to rewrite both fractions over a single denominator.
• This step simplifies the overall structure, making further reduction easier.

Once achieved, you are free to combine terms across the numerator and denominator efficiently.

Factoring

Factoring plays a critical role in simplifying complex rational expressions. After rewriting the expression using a common denominator, the next step is to factor wherever possible.

In our exercise, after combining fractions and simplifying, both the numerator and the denominator need to be factored:\[yx^2 - xy^2 - x^2 y - x^3 \quad \text{and}\quad x^3 - x^2 y + xy^2 - y^3.\]By factoring out common terms, you can further simplify the expression. In this case, you notice that both the numerator and the denominator share \((x+y)\) as a factor:

• Factor out the common elements from both the numerator and the denominator.
• This reveals terms which can potentially be cancelled out.
• Continue to factor further wherever possible to reduce the expression to its simplest form.

Through factoring, you can significantly reduce the complexity of the expression, allowing for easy cancellation of terms and ultimately leading to the simplified final form.