In this example, finding the common denominator waspretty easy . The common
denominator is 6. We found it by multiplying the original denominators, 2
Notice that 2 and 3 are both prime numbers.
What happens if the beginning denominators are not prime numbers?
We could find a common denominator as we did before: just
multiply 6 and 15 together:
This involved a lot of work! Also, although it may not be
obvious, the final answer is
not in lowest terms; the fraction can
We can save ourselves from working with larger numbers by
Common Denominator (LCD). The process for finding the LCD is outlined below;
before we look at it, consider this: The "LCD process" is more complicated than
multiplying denominators together, and seems hard when you first learn it; but
worth the effort. Here are its advantages:
Here's an example: we will find the LCD of the following
fractions (we won't bother to
carry out the addition of the fractions in this example).
Ex: Given and
. What is the LCD of these two fractions?
Step 1. Factor 24 and 180, using exponential notation:
Step 2. List all the different prime numbers in
2, 3, 5 are all the prime numbers that occurred.
(Notice that 2 and 3 occurred in both factorizations, and 5
occurred only in the second factorization.)
Step 3. Raise each prime number to its highest power, and multiply:
The first prime is the number 2. It's raised to the third power in the first
factorization, and it's raised to the second power in the second factorization.
highest power is 3. Write .
The second prime is 3. It's raised to the first power (so no exponent is
= 3) in the first factorization, and it's raised to the second power in the
factorization. So its highest power is 2. Write
The third prime is 5. It's not in the first factorization, and it's raised to
power in the second factorization . So its
highest power is 1, which we don't
need to write. We write only 5.
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