
äußern
to express
To put thoughts, feelings, opinions, or wishes into words and communicate them.
Er hat den Wunsch geäußert, früher nach Hause zu gehen.
He expressed the wish to go home earlier.
to express
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to express
To put thoughts, feelings, opinions, or wishes into words and communicate them.
Er hat den Wunsch geäußert, früher nach Hause zu gehen.
He expressed the wish to go home earlier.

to comment
To take a position on a particular topic or say one's opinion about it.
Der Politiker wollte sich nicht zu den aktuellen Vorwürfen äußern.
The politician did not want to comment on the current accusations.

to manifest
To show itself in the form of symptoms or signs; to manifest itself (mainly in medicine).
Die Krankheit äußert sich durch hohes Fieber und starken Husten.
The illness manifests itself through a high fever and a severe cough.
The verb 'äußern' (to express) is related to the adverb 'außen' (outside). It describes the action of bringing thoughts or feelings from the inside to the outside, that is, making them visible or audible.
Er äußert selten seine persönliche Meinung in der Gruppe.
He rarely expresses his personal opinion in the group.
This verb is very often used reflexively as 'sich äußern' (to comment / to express oneself). Make sure to use the correct reflexive pronoun (mich, dich, sich, uns, euch, sich).
Ich möchte mich dazu jetzt nicht äußern.
I don't want to comment on that now.
In this meaning, 'äußern' (to express) is often combined with nouns that describe an opinion or feeling. Typical combinations are: Kritik äußern (to express criticism), Bedenken äußern (to express concerns), einen Verdacht äußern (to voice a suspicion), or eine Vermutung äußern (to express an assumption).
Mehrere Mitarbeiter äußerten Bedenken gegen den neuen Plan.
Several employees expressed concerns about the new plan.
When taking a position on a topic, you use the fixed expression 'sich zu etwas äußern' (to comment on something). The preposition 'zu' always takes the dative here.
Die Ministerin wird sich morgen zu den Gerüchten äußern.
The minister will comment on the rumors tomorrow.
This meaning is typical of medicine or science. The subject of the sentence is usually an illness, a symptom, or a problem that becomes visible.
Eine Allergie kann sich durch Hautausschlag äußern.
An allergy can manifest itself through a skin rash.
To describe the type of symptoms, the prepositions 'durch' (with accusative) or 'in' (with dative) are used. Both are possible and have a similar meaning.
Seine Nervosität äußerte sich durch zitternde Hände.
His nervousness manifested itself through trembling hands.
to show
Showing a feeling means that it becomes recognizable through behavior, facial expression, or a reaction. Expressing a feeling usually means putting it explicitly into words and communicating it. So zeigen is appropriate for visible behavior, while äußern is appropriate for a verbal communication.

Sie äußerte ihren Ärger mit deutlichen Worten.
She expressed her anger in clear words.
Sie zeigte ihren Ärger durch ihren ernsten Blick.
She showed her anger through her serious expression.