Add Rational Expressions With Unlike Denominators

Add rational expressions with unlike denominators

When two rational expressions have different denominators, you cannot add them directly. First rewrite both expressions so they share a common denominator, then combine the numerators.

Steps

1. Find a common denominator. Factor the denominators if possible, then identify the least common denominator (LCD) or any common denominator that works.
2. Rewrite each fraction. Multiply the numerator and denominator of each expression by whatever factor is missing to reach the common denominator.
3. Add the numerators. Keep the common denominator the same and combine only the numerators.
4. Simplify the result. Factor the numerator if possible and reduce any common factors with the denominator.

Important check

Make sure no denominator becomes zero for the values involved, and verify that both original expressions were rewritten correctly. If the final numerator and denominator share a factor, reduce the fraction.

Example of the process

If the denominators are unlike, such as two different linear factors, look for the product or a smaller shared factor as the common denominator. The goal is to express both parts over the same base, then add and simplify.