ease / lightness
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ease
The quality of being simple and not involving great effort or difficulty.
Er löste die komplexe mathematische Aufgabe mit einer erstaunlichen Leichtigkeit.
He solved the complex mathematical problem with astonishing ease.
The word is derived from the adjective 'leicht' (light/easy) and turned into a noun with the ending '-igkeit'. This ending often forms abstract nouns from adjectives, and the meanings of 'Leichtigkeit' (ease/lightness) correspond directly to the meanings of 'leicht' (easy, carefree, low in weight).
Die Leichtigkeit des neuen Materials ist beeindruckend.
The lightness of the new material is impressive.
You can recognize the meaning from the context: If it is about an action, 'Einfachheit' (simplicity/ease) is meant. If it is about feelings or an atmosphere, 'Unbeschwertheit' (carefreeness) is meant. If it is about a physical object, the low 'Gewicht' (weight) is meant.
Er erledigte die Arbeit mit großer Leichtigkeit. (Handlung)
He completed the work with great ease. (action)
In the meaning of 'einfach' (easy), the word is very often used in the fixed expression 'mit Leichtigkeit' (with ease). This combination expresses that someone does something without effort.
Die Turnerin absolvierte die schwierige Übung mit scheinbarer Leichtigkeit.
The gymnast completed the difficult exercise with apparent ease.
In the sense of 'Unbeschwertheit' (carefreeness), the word is often combined with verbs of feeling and enjoyment. A well-known philosophical expression is 'die Leichtigkeit des Seins' (the lightness of being).
Im Urlaub fand sie zu ihrer alten Leichtigkeit zurück.
On vacation, she found her old lightness again.
effort / trouble
'die Mühe' and 'die Leichtigkeit' are opposites here. 'Mühe' means that something takes effort or is hard to manage. 'Leichtigkeit' means that something goes easily and takes hardly any strength or causes hardly any problems. You use 'Mühe' when a task is difficult. 'Leichtigkeit' fits when something succeeds almost without effort. So the contrast is: with a lot of effort versus without much effort.
Lisa erklärt das Thema mit einer Leichtigkeit, die alle verstehen.
Lisa explains the topic with an ease that everyone understands.
Er löst die Aufgabe nur mit großer Mühe.
He solves the task only with great effort.