key term - Least Common Denominator

Definition

The least common denominator (LCD) is the smallest positive integer that is a multiple of all the denominators in a set of fractions. It is a crucial concept in working with fractions, adding and subtracting rational expressions, and solving rational equations.

5 Must Know Facts For Your Next Test

1. The least common denominator is the smallest positive integer that is divisible by all the denominators in a set of fractions.
2. Finding the LCD is a necessary step when adding, subtracting, or comparing fractions with different denominators.
3. To find the LCD, you must first list all the denominators, then find the prime factorization of each denominator, and finally take the product of the highest power of each prime factor.
4. The LCD is used to convert fractions to equivalent fractions with the same denominator, which is a crucial step in performing operations with rational expressions.
5. Solving rational equations often requires finding the LCD of the denominators in order to eliminate the denominators and solve for the variable.

Review Questions

• Explain the purpose of finding the least common denominator when working with fractions.
  • The least common denominator (LCD) is crucial when working with fractions because it allows you to convert fractions with different denominators into equivalent fractions with a common denominator. This is necessary for performing operations like addition, subtraction, and comparison of fractions. By finding the LCD, you can ensure that all the fractions have the same denominator, which makes the calculations much easier and more straightforward.
• Describe how the least common denominator is used in the context of adding and subtracting rational expressions.
  • When adding or subtracting rational expressions, the least common denominator (LCD) is used to convert the expressions to have a common denominator. This is done by finding the LCD of all the denominators in the given rational expressions. Once the expressions have a common denominator, you can then add or subtract the numerators and write the result over the common denominator. The LCD ensures that the operations are performed correctly and that the resulting expression is in its simplest form.
• Analyze the role of the least common denominator in solving rational equations.
  • In solving rational equations, the least common denominator (LCD) is essential for eliminating the denominators and isolating the variable. By finding the LCD of all the denominators in the equation, you can convert the rational expressions to have a common denominator. This allows you to clear the denominators by multiplying both sides of the equation by the LCD, effectively removing the fractions and leaving you with a polynomial equation that can be solved using standard algebraic techniques. The LCD is a crucial step in the process of solving rational equations.

Related terms

Fraction:

A representation of a part of a whole, expressed as a numerator over a denominator.

Rational Expression:

A quotient of two polynomials, where the denominator is not equal to zero.

Rational Equation:

An equation that contains one or more rational expressions.