
der Kauf
purchase
The process of acquiring something in exchange for payment; the act of buying.
Der Kauf des Hauses muss notariell beglaubigt werden.
The purchase of the house must be notarized.
purchase
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purchase
The process of acquiring something in exchange for payment; the act of buying.
Der Kauf des Hauses muss notariell beglaubigt werden.
The purchase of the house must be notarized.

purchase
The thing that was bought; the acquired item.
Diese Jacke war ein wirklich guter Kauf, sie war stark reduziert.
This jacket was a really good purchase; it was heavily reduced.

to accept
To consciously accept a disadvantage or inconvenience in order to achieve a desired goal.
Für seinen Traumjob nahm er auch einen längeren Arbeitsweg in Kauf.
For his dream job, he also accepted a longer commute.
'Der Kauf' (the purchase) is the noun corresponding to the verb 'kaufen' (to buy). It describes either the act of buying or the thing that was bought.
Wir müssen den Kauf der neuen Waschmaschine gut planen.
We need to plan the purchase of the new washing machine carefully.
To say what is being bought, German often uses the genitive or the preposition 'von' with the dative. The genitive is more formal.
Der Kauf des alten Hauses war eine große Entscheidung.
The purchase of the old house was a big decision.
When 'der Kauf' (the purchase) refers to the thing that was bought, it is often used with evaluative adjectives such as 'ein guter' (a good), 'ein schlechter' (a bad), 'ein teurer' (an expensive), or 'ein günstiger Kauf' (a cheap/good-value purchase).
Das Fahrrad für nur 100 Euro war ein wirklich günstiger Kauf.
The bike for only 100 euros was a really good-value purchase.
The expression 'etwas in Kauf nehmen' means to consciously accept a disadvantage in order to achieve a goal. The disadvantage that is accepted is in the accusative.
Um am Meer zu wohnen, nehme ich die hohe Miete in Kauf.
In order to live by the sea, I accept the high rent.
takeover / assumption
When a company is bought, the paid transaction is the focus. Übernahme, by contrast, refers to the fact that one company thereby gains control over another. When the price and the purchase transaction are at issue, Kauf is appropriate; when the change of control is the focus, Übernahme is more suitable.

Der Kauf der Firma kostete den Konzern zehn Millionen Euro.
The purchase of the company cost the group ten million euros.
Nach der Übernahme kontrolliert der Konzern alle Entscheidungen der Firma.
After the takeover, the group controls all of the company's decisions.