Finding The Least Common Denominator for Fractions

To find the LCD, Least Common Denominator, for two or more fractions

• Factor each denominator as a product of prime factors.
• Express each factorization in exponential form.
• List all the distinct prime factors found in the factorizations.
• The LCD is a product of powers of these distinct prime factors where the exponent on any prime factor is the highest power of that prime factor that appears in any factorization.

Examples:

Find the lcd for the fractions:  2/15,  5/12 and 7/18
15 = 3(5)     12 = 2² (3)     18 = 3² (2)
The three distinct factors appearing in the three factorizations are 2, 3, and 5.
The highest power of two appearing is the second power.
The highest power of three appearing is the second power.
The highest power of five appearing is the first power,
The LCD is 2² (3² )(5).

Find the LCD for the fractions:  2/(x²  +2x + 1) and 5/(x²  - 1)
x²  +2x + 1 = (x + 1)²     x²  - 1 = x + 1)(x - 1)
The two distinct factors appearing in the factorizations are x + 1 and x - 1.
The highest power of x + 1 is the second power.
The highest power of x - 1 is the first power.
The LCD is (X + 1)² (x - 1).

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