Adding rational expressions with a common denominator means combining their numerators while retaining the shared denominator: , where . The learner interprets the expressions as quantities partitioned into equal algebraic units, simplifies the resulting numerator when appropriate, and preserves restrictions on variable values; adding denominators or canceling terms across addition produces incorrect results. This understanding supports simplifying rational equations and functions.
When rational expressions have the same denominator, keep that denominator and add the numerators:
The denominator tells you the size of each algebraic unit, so it is not added. First note any restriction: the denominator cannot equal zero.
Example:
Step 1: State the restriction.
The denominator is , so
Step 2: Keep the common denominator.
Since both expressions have denominator , write one fraction with that denominator:
Step 3: Combine like terms in the numerator.
Therefore,
Do not add the denominators. For example, the denominator is not . Also, do not cancel terms across addition; simplify only after combining the numerators, and keep the restriction .
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