Problem 91 - College Algebra

Question

Explain how to add or subtract rational expressions with the same denominators.

Step-by-Step Solution

Answer

The procedure for adding or subtracting rational expressions with the same denominator is quite straightforward. It all boils down to performing addition or subtraction operation on the numerators of these expressions while the common denominator remains unchanged. Simplifying the resulting expression concludes the process.

Step 1: Identify Common Denominator

The first step is to identify that the rational expressions have the same denominator, this is essential when adding or subtracting these expressions. Suppose we have the rational expressions \(\frac{a}{c}\) and \(\frac{b}{c}\). Both expressions have the same denominator, \(c\). 'a' and 'b' represent the numerators of the rational expressions and are what we're primarily concerned with for addition or subtraction.

Step 2: Add or Subtract Numerators

The second step is to perform the addition or subtraction operation on the numerators of these rational expressions. To add, it looks like this: \(\frac{a}{c} + \(\frac{b}{c} = \frac{a+b}{c}\); to subtract: \(\frac{a}{c} - \(\frac{b}{c} = \frac{a-b}{c}\). It is that simple. You are adding or subtracting the numerators while keeping the denominator constant.

Step 3: Simplify Resulting Expression

The final step is to simplify the resulting expression, if possible. It may be that the result cannot be simplified, in which case, you're done. However, if there is room for simplification, e.g., \( \frac{4+2}{6} \) simplifies to \( \frac{6}{6} = 1 \). This could involve factoring the numerator and denominator and cancelling common factors.