pāpakaiḥ

 The Sanskrit term "pāpakaiḥ" is the instrumental plural 

form of "pāpaka," meaning "by the wicked," "by evil-doers," 

or "through sinful persons."

Detailed Explanation

Root word: pāpaka (पापक)

• Derived from pāpa (पाप), meaning "sin," "evil," or "wrongdoing."
• The suffix -ka indicates an agent or one associated with 
• a quality, so pāpaka generally means "wicked," "evil-doer," 
• or "one who commits sin."

Grammatical form: pāpakaiḥ (पापकैः)

• Case: Instrumental (tṛtīyā vibhakti)
• Number: Plural
• Meaning in context: "by the wicked," "through evil-doers," 
• or "with those who are sinful."
• The instrumental case is typically used in Sanskrit to indicate 
• agency or the instrument by which an action is performed.

Example usage in a sentence:

• सत्यं धर्ममार्गेण रक्षितं भवति pāpakaiḥ न भविष्यति।
  Translation: "Truth and righteousness are protected on the path 
• of dharma; they are not harmed by the wicked."

In summary, pāpakaiḥ conveys the sense of action done “by” 

or “through” multiple wicked or sinful persons and is commonly 

found in classical Sanskrit literature and philosophical texts.