Problem 15 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{\frac{a}{b}+\frac{c}{b}}{\frac{a}{b}-\frac{c}{b}} $$

Step-by-Step Solution

Answer

$$ \frac{\frac{a}{b}+\frac{c}{b}}{\frac{a}{b}-\frac{c}{b}} $$ Answer: The simplified form of the complex rational expression is: $$ \frac{a+c}{a-c} $$

Step 1: Identify the common denominators for the fractions in the numerator and the denominator

In both the numerator and the denominator, there are two fractions with the denominator 'b'. Therefore, the common denominator is simply 'b'.

Step 2: Clear the fractions in the numerator and the denominator

To clear the fractions in the numerator and the denominator, multiply both by 'b'. $$ \frac{\frac{a}{b}+\frac{c}{b}}{\frac{a}{b}-\frac{c}{b}} \cdot \frac{b}{b} = \frac{a+c}{a-c} $$

Step 3: Simplify the expression, if possible

Based on the process, the expression is already simplified: $$ \frac{a+c}{a-c} $$ So the simplified complex rational expression is: $$ \frac{a+c}{a-c} $$

Key Concepts

Simplifying Expressions

When tackling complex rational expressions, the key idea is to break down the expression into simpler parts. The term 'simplifying expressions' refers to the process of reducing complexity by performing operations that lead to a more straightforward form. Complex rational expressions often feature fractions in both the numerator and the denominator. The goal here is to reduce this into a single fraction.

• First, identify any common terms or factors that can be canceled or simplified across the fractions.
• Second, perform necessary arithmetic operations to combine and then reduce the expressions.

In the end, a successful simplification should leave you with a clean and efficient expression that captures the essence of the original, but in an easier-to-manage form.

Common Denominator

Finding a common denominator is essential when working with complex rational expressions as it allows you to combine fractions within the numerator and denominator effectively. In our exercise, the fractions alternate between the numerators and denominators but share a common denominator 'b'.

• Identify the least common denominator (LCD) in both the numerator and the denominator fractions. The LCD is the smallest shared denominator that allows each fraction involved to be expressed uniformly.
• Transform each fraction so that their denominators match the common denominator before attempting to perform addition or subtraction.

This step sets the stage for eliminating fractions in later steps.

Fractions in Numerator/Denominator

Dealing with fractions in both the numerator and the denominator can be tricky, but it's manageable with systematic strategies. When fractions appear both above and below the division line of a complex expression, the first task is clearing those fractions by using multiplication. In this process:

• Multiply both the entire numerator and the entire denominator by the common denominator found previously. This operation helps clear the smaller fractions by effectively multiplying away their denominators.
• This step simplifies the expression within each level of the fraction, leading to easily manageable algebraic expressions rather than fractional components.

By the end, working with a single simplified complex fraction rather than two nested fractions becomes possible.

Algebraic Fractions

Algebraic fractions, much like numerical fractions, need to be handled with specific algebraic rules, especially when simplifying complex rational expressions. These types of expressions have polynomials within their numerator and/or denominator. Key strategies include:

• Factor polynomials fully whenever possible. This creates possibilities to cancel out common factors between the numerator and the denominator.
• Remember to check for restrictions since division by zero is undefined. Look for values of variables that might lead to a zero in the denominator and treat them appropriately in your solutions.

Being attentive to these details ensures that your simplified form is precise and mathematically accurate, providing a correct solution that abides by algebraic principles.