8.4 Add and Subtract Rational Expressions with Unlike Denominators

Rational expressions with unlike denominators can be tricky, but they're essential for algebra. You'll learn how to find the least common denominator and use it to add or subtract fractions. This skill is crucial for solving more complex equations later on.

Don't worry if it seems complicated at first. With practice, you'll get the hang of converting fractions, adding numerators, and simplifying results. These techniques will help you tackle more advanced math problems in the future.

Adding and Subtracting Rational Expressions with Unlike Denominators

Least common denominator for rational expressions

• Find the smallest common multiple of the denominators called the least common denominator (LCD)
  • Determine the prime factorization of each denominator
  • LCD is the product of the highest power of each prime factor among all denominators
• Example: Find the LCD of $\frac{2}{15}$ and $\frac{3}{20}$
  • Prime factorization of 15: $3 \times 5$
  • Prime factorization of 20: $2^2 \times 5$
  • Highest power of 2 is 2, highest power of 3 is 1, and highest power of 5 is 1
  • LCD = $2^2 \times 3 \times 5 = 60$

Conversion to common denominators

• Multiply the numerator and denominator of each expression by the appropriate factor to convert rational expressions to equivalent forms with a common denominator
  • Appropriate factor is the LCD divided by the current denominator
• Example: Convert $\frac{2}{15}$ and $\frac{3}{20}$ to equivalent forms with the LCD of 60
  • For $\frac{2}{15}$, multiply numerator and denominator by $\frac{60}{15} = 4$: $\frac{2}{15} \times \frac{4}{4} = \frac{8}{60}$
  • For $\frac{3}{20}$, multiply numerator and denominator by $\frac{60}{20} = 3$: $\frac{3}{20} \times \frac{3}{3} = \frac{9}{60}$
• This process involves cross multiplication to ensure the fractions remain equivalent

Addition of unlike rational expressions

• Convert rational expressions with different denominators to equivalent forms with a common denominator (the LCD)
• Add the numerators and keep the common denominator
• Simplify the result if possible by reducing the fraction
• Example: Add $\frac{2}{15} + \frac{3}{20}$
  1. Convert to equivalent forms with LCD of 60: $\frac{8}{60} + \frac{9}{60}$
  2. Add numerators: $\frac{8+9}{60} = \frac{17}{60}$
  3. Simplify the result: $\frac{17}{60}$ cannot be reduced further

Subtraction of unlike rational expressions

• Convert rational expressions with different denominators to equivalent forms with a common denominator (the LCD)
• Subtract the numerators and keep the common denominator
• Simplify the result if possible by reducing the fraction
• Example: Subtract $\frac{7}{12} - \frac{5}{18}$
  1. Convert to equivalent forms with LCD of 36: $\frac{21}{36} - \frac{10}{36}$
  2. Subtract numerators: $\frac{21-10}{36} = \frac{11}{36}$
  3. Simplify the result: $\frac{11}{36}$ cannot be reduced further

Working with algebraic fractions

• Algebraic fractions are rational expressions with variables in the numerator, denominator, or both
• When adding or subtracting algebraic fractions, factoring may be necessary to find the LCD
• Simplification of the result often involves factoring to cancel common terms
• Complex fractions (fractions containing fractions) may require additional steps to simplify before adding or subtracting