Explain the steps involved in adding two rational expressions.
Understand the Problem
The question is asking for a detailed explanation of the process for adding two rational expressions, which typically involves finding a common denominator, combining the numerators, and simplifying the result if possible.
Answer
Add the two rational expressions and simplify the result.
Answer for screen readers
The final answer is the sum of the two rational expressions written as a single fraction with the common denominator.
Steps to Solve
- Identify the denominators of the rational expressions
First, look at the denominators (the bottom part of the fractions) of the two rational expressions. For example, if we have $$\frac{a}{b}$$ and $$\frac{c}{d}$$, the denominators are $$b$$ and $$d$$.
- Find the least common denominator (LCD)
The least common denominator is the smallest number that is divisible by both denominators. For the denominators $$b$$ and $$d$$, we find their LCD, which we'll call $$L$$.
- Rewrite each fraction with the LCD as the new denominator
Convert each rational expression so that they both have the LCD as their new denominator. To do this, multiply the numerator and denominator of each rational expression by whatever value makes the denominator equal to the LCD.
For example, if the LCD is $$L$$: $$\frac{a}{b} = \frac{a \cdot \frac{L}{b}}{b \cdot \frac{L}{b}} = \frac{a \cdot \frac{L}{b}}{L}$$ $$\frac{c}{d} = \frac{c \cdot \frac{L}{d}}{d \cdot \frac{L}{d}} = \frac{c \cdot \frac{L}{d}}{L}$$
- Combine the numerators over the common denominator
Once both rational expressions have the same denominator, add their numerators together to create a new rational expression.
$$\frac{a \cdot \frac{L}{b} + c \cdot \frac{L}{d}}{L}$$
- Simplify the resulting rational expression
If possible, simplify the expression by reducing the common factors in the numerator and denominator. This simplification step may involve factoring and canceling out common terms.
The final answer is the sum of the two rational expressions written as a single fraction with the common denominator.
More Information
Rational expressions are added similarly to numeric fractions, requiring a common denominator to combine them.
Tips
A common mistake is forgetting to find the least common denominator and attempting to add the fractions directly. Another mistake is not fully simplifying the final expression.
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