Adding Rational Expressions (Unlike Denominators)
Learn how to add rational expressions with different denominators by finding the LCD.
Definition
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Worked Examples
Simplify:
Identify the denominators
Denominator 1: , Denominator 2: → Both are linear, no common factors
Find the LCD
LCD = → Multiply the distinct factors
Rewrite first fraction
→
Rewrite second fraction
→
Add the numerators
→
Answer: or
Common Mistakes
Forgetting to factor denominators first
Why it's wrong: Without factoring, you might miss common factors and create an LCD that is larger than necessary, making the problem harder.
Correct: Always factor each denominator completely before finding the LCD. For example, .
Adding denominators instead of finding LCD
Why it's wrong: Unlike numerators, denominators are not added together. The LCD must contain all factors from each denominator.
Correct: . The LCD is .
Forgetting to multiply both numerator and denominator
Why it's wrong: When converting to the LCD, you must multiply by a form of 1 (same expression over itself) to keep the value unchanged.
Correct: , not .
Not simplifying the final answer
Why it's wrong: After adding, the resulting numerator might share a common factor with the denominator.
Correct: Always check if the numerator can be factored and if any factors cancel with the denominator.
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Practice Problems
17 problemsWhat is the LCD of and ?
Why It Matters
- Physics: Combining resistances in parallel circuits:
- Work problems: If one worker completes a job in hours and another in hours, together they complete of the job per hour
- Optics: The thin lens equation:
- Rate problems: Combining different rates of production or travel
Real World Applications
Parallel Resistors in Electronics
When resistors are connected in parallel, the total resistance is found using the formula with unlike denominators.
Example:
For two resistors of ohms and ohms in parallel:
You have two resistors: one with resistance ohms and another with resistance ohms connected in parallel.
Find an expression for the total resistance.
Step 1: Write the mathematical expression
First find
Combined Work Rate Problems
When two people work together, their combined rate involves adding fractions with different denominators.
Example:
If Worker A completes a job in hours and Worker B in hours, their combined rate is jobs per hour.
Machine A produces a batch in hours. Machine B produces the same batch in hours.
What fraction of the batch do both machines produce together in one hour?
Step 1: Write the mathematical expression
Combined rate =
Key Takeaways
- 1To add rational expressions with unlike denominators, first factor all denominators completely
- 2Find the LCD by including each factor the maximum number of times it appears in any denominator
- 3Multiply each fraction by a form of 1 to convert to the LCD
- 4Add the numerators while keeping the common denominator
- 5Simplify the result by factoring and canceling common factors
Frequently Asked Questions
Glossary
- Rational expression
- A fraction where the numerator and/or denominator contains a polynomial
- LCD (Least Common Denominator)
- The smallest expression that is divisible by all denominators in a problem
- Unlike denominators
- Denominators that are not identical and require finding a common denominator
- Factor
- To write an expression as a product of simpler expressions
Formula Card
Adding Two Rational Expressions
General formula when denominators have no common factors
LCD Method
Rewrite each fraction using the Least Common Denominator