52.7 Rational Operation Sequences

Rational Operation Sequences are expressions that combine more than one of the four rational operations, multiplication, division, addition, and subtraction, in a single problem, requiring the individual operation procedures to be applied in the correct order while tracking domain restrictions across every stage rather than only at the very end. Because each individual operation, multiplication, division, addition, or subtraction, has already been defined on its own, working through a sequence is primarily a matter of applying the standard order of operations to decide which individual procedure to perform first, then carrying the result forward into the next stage.

This topic emphasizes that domain restrictions accumulate throughout a multi-step sequence, since every rational expression that appears at any stage, not just the ones present in the final written form, can contribute a restriction that must be included in the complete final answer.

Starting the Sequence

Initial Restriction Collection

Before combining any expressions, the individual domain restriction of every rational expression appearing anywhere in the original sequence is identified, exactly as it would be for a single rational expression, so that no restriction is overlooked once the expressions begin merging together.

Parenthesized Rational Subexpression

If any part of the sequence is grouped in parentheses, that grouped subexpression is treated as its own self-contained operation and is fully simplified first, before its result is combined with the rest of the sequence, following the same grouping-first priority used in any order-of-operations problem.

Applying the Standard Order

Rational Product-or-Quotient Priority

Following the standard order of operations, any multiplication or division present in the sequence is carried out before any addition or subtraction, applying the rational multiplication or rational division procedure to the relevant pair of expressions first.

Rational Sum-or-Difference Stage

Once every multiplication or division in the sequence has been resolved, any remaining addition or subtraction is carried out using the rational addition or rational subtraction procedure, constructing an LCD across whatever expressions remain at this stage.

Managing Multi-Step Results

Intermediate Expression Factoring

After each individual operation within the sequence is completed, the resulting intermediate expression is factored where possible, both to prepare it for the next operation in the sequence and to reveal any cancellation opportunities before they become harder to spot in a more complex combined expression.

Intermediate Domain Carryover

Every domain restriction collected from the original expressions continues to apply to every intermediate result throughout the sequence, and the running list of restrictions is carried forward unchanged from one stage to the next, regardless of whether a particular restricted value's originating factor is still visibly present in the current intermediate expression.

Completing the Sequence

Final Rational Reduction

Once every operation in the sequence has been performed in the correct order, the final combined expression is factored and reduced one last time, cancelling any remaining shared factors between its numerator and denominator, and the complete accumulated set of domain restrictions from every stage is stated alongside this final simplified result.