
reagieren auf
to react to
To respond with an action or statement to an action, statement, or event.
Er hat auf meine E-Mail immer noch nicht reagiert.
He still hasn't reacted to my email.
to react
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to react to
To respond with an action or statement to an action, statement, or event.
Er hat auf meine E-Mail immer noch nicht reagiert.
He still hasn't reacted to my email.

to react
In chemistry or physics, to interact with another substance and change in the process.
Natrium reagiert heftig, wenn es mit Wasser in Kontakt kommt.
Sodium reacts violently when it comes into contact with water.

to react to
To show a physical reaction, often an allergic one, to a particular substance or stimulus.
Mein Körper reagiert allergisch auf Pollen.
My body reacts allergically to pollen.
The verb 'reagieren' (to react) is almost always used with the preposition 'auf' and the accusative. You should memorize this pattern as a fixed unit: 'reagieren auf etwas/jemanden' (to react to something/someone).
Wie hat sie auf die Nachricht reagiert?
How did she react to the news?
Verbs that end in '-ieren' are regular and form the past participle without 'ge-'. So it is 'hat reagiert' (has reacted), not 'hat gereagiert'. This is a useful rule for many verbs of Latin origin.
Ich habe sofort auf seine Frage reagiert.
I reacted immediately to his question.
The context makes clear which meaning is intended. If it is about a person or a message, it refers to a social reaction. If it is about substances such as pollen or nuts, it refers to a physical reaction. In a scientific context, it refers to a chemical reaction.
Sie reagierte nicht auf seine Entschuldigung.
She did not react to his apology.
To describe the type of reaction, 'reagieren' (to react) is often combined with adverbs. Typical examples are 'schnell' (quickly), 'gelassen' (calmly), 'überrascht' (with surprise), 'wütend' (angrily), or 'gar nicht' (not at all) reagieren.
Er hat sehr gelassen auf die Kritik reagiert.
He reacted very calmly to the criticism.
to answer
Both words can refer to a response to an action, statement, or feeling. „Beantworten“ emphasizes a corresponding counteraction and takes its target directly, while „reagieren“ is more general and is usually used with „auf“. Use „beantworten“ for a specific reply and „reagieren“ for any kind of reaction.

Sie reagierte auf seinen freundlichen Gruß mit einem Lächeln.
She reacted to his friendly greeting with a smile.
Sie beantwortete seinen freundlichen Gruß mit einem Lächeln.
She returned his friendly greeting with a smile.