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Difference between Make out a prescription and Make up a prescription

make out a prescription—(of a doctor) fill out a written order for a medicine:

  • Many over-the-counter drugs are very difficult to get in other countries, so have your doctor make out a prescription as a precaution.

make up a prescription—(of a chemist) prepare a mixture such as medicine:

  • Physicians are not allowed to sell medicines to their patients, except when the pharmacist refuses to make up a prescription.