Problem 14 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{a+\frac{1}{x}}{a-\frac{1}{x}} $$

Step-by-Step Solution

Answer

Question: Simplify the complex rational expression: \(\frac{a + \frac{1}{x}}{a - \frac{1}{x}}\). Answer: \(\frac{ax + 1}{ax - 1}\)

Step 1: Identify the common denominator

The given expression has two fractions: \(\frac{1}{x}\) in the numerator and \(\frac{-1}{x}\) in the denominator. The common denominator for these two fractions is \(x\).

Step 2: Multiply the numerator and the denominator by the common denominator

To eliminate the fractions within the expression, we multiply both the numerator and the denominator by the common denominator \(x\). $$ \frac{a + \frac{1}{x}}{a - \frac{1}{x}} \cdot \frac{x}{x} $$

Step 3: Distribute the common denominator

Now, distribute \(x\) to each term in the numerator and the denominator. $$ \frac{(a+\frac{1}{x})\cdot x}{(a-\frac{1}{x})\cdot x} = \frac{ax + (\frac{1}{x})\cdot x}{ax - (\frac{1}{x})\cdot x} $$

Step 4: Simplify the expression

After distributing the common denominator, we'll simplify the numerator and denominator as follows: $$ \frac{ax + (\frac{1}{x})\cdot x}{ax - (\frac{1}{x})\cdot x} = \frac{ax + 1}{ax - 1} $$ The complex rational expression has been simplified to \(\frac{ax + 1}{ax - 1}\).

Key Concepts

Common Denominator

Understanding the concept of a common denominator is crucial when simplifying complex rational expressions. A common denominator refers to a shared multiple of the denominators of two or more fractions. By identifying a common denominator, we can combine fractions more easily.

For instance, with the expression \[\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\], we have a fraction in the numerator and another in the denominator. Here, both interior fractions have the same denominator, which is \(x\). The common denominator is thus that exact value of \(x\). Incorporating the insights of a common denominator can dramatically streamline the process of working with complex fractions.

Rational Expression Simplification

Simplification of rational expressions is a process in algebra that aims to make expressions easier to understand and work with. Simplifying can involve reducing fractions to their lowest terms, combining like terms, or eliminating complex fractions by finding a common denominator.

In the exercise \[\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\], simplification begins by finding a common denominator, and then multiplying both parts of the complex fraction by it. This eliminates the smaller fractions within the larger fraction, leading to a much simpler rational expression. It’s akin to clearing out the unnecessary intricacies to reveal a more straightforward form.

Distributing the Common Denominator

Distributing the common denominator is a step that comes right after identifying it. In our example, after determining that \(x\) is our common denominator, we distribute it across the fraction — both to the numerator and denominator.

This process involves multiplying each term of the numerator and the denominator by the common denominator. It is shown as follows:\[\frac{(a+\frac{1}{x})\times x}{(a-\frac{1}{x})\times x}\].

Distributing the common denominator through multiplication helps to make the complex fraction much simpler, leading to an easier simplification of the entire expression. This is a vital step in the algebraic process of dealing with rational expressions.

Algebraic Fractions

Algebraic fractions are fractions that contain polynomials in their numerators, denominators, or both. They are akin to numerical fractions but involve variables. Simplifying these types of fractions often requires the same techniques used for numeric fractions, but also includes finding common denominators and simplifying variables.

For instance, in the exercise\[\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\], the entire structure is an algebraic fraction. After distributing the common denominator and simplification, we are left with a more manageable algebraic fraction \(\frac{ax+1}{ax-1}\). Hence, algebraic fractions, once simplified, can be as easy to work with as their numeric counterparts.