why might it be important for a drug to have a brief half-life when it has a narrow therapeutic-index?

Answer:
A brief half life helps prevent dangerous doses from building up whilst using drugs where the treatment dose is very close to a potentially dangerous one.

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A narrow therapeutic index means that the difference in dosage between therapeutic and toxic levels is very small, and so must be monitored carefully because the patient can easily become toxic. A brief half-life is important because the shorter the half-life is, the faster it will be cleared from the system -- thus, if the patient reaches toxic levels of the drug, the patient will be able to recover faster if the drug has a short half-life than if it had a long one.

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