mathematisch
mathematical
Relating to the science of mathematics, based on it, or working with its methods.
Er löste die Aufgabe mit einer komplexen mathematischen Formel.
He solved the problem with a complex mathematical formula.

mathematical
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mathematical
Relating to the science of mathematics, based on it, or working with its methods.
Er löste die Aufgabe mit einer komplexen mathematischen Formel.
He solved the problem with a complex mathematical formula.

The adjective 'mathematisch' (mathematical) is derived directly from the noun 'die Mathematik' (mathematics). The ending '-isch' is a typical ending used to form adjectives from nouns.
Für diesen Studiengang sind gute mathematische Kenntnisse erforderlich.
Good mathematical knowledge is required for this degree program.
In its literal meaning, 'mathematisch' (mathematical) is often combined with terms from science. Typical examples are 'eine mathematische Formel' (a mathematical formula), 'ein mathematisches Problem' (a mathematical problem), or 'ein mathematischer Beweis' (a mathematical proof).
Der Professor erklärte uns das mathematische Modell hinter seiner Theorie.
The professor explained the mathematical model behind his theory to us.
You can tell whether the literal or figurative meaning is intended from the noun. If it refers to an abstract concept like 'Genauigkeit' (accuracy) or 'Präzision' (precision), the figurative meaning is intended.
Der Uhrmacher arbeitet mit mathematischer Präzision.
The watchmaker works with mathematical precision.
In the figurative meaning, 'mathematisch' (mathematical) describes extreme exactness. Common combinations are 'mathematische Genauigkeit' (mathematical accuracy), 'mathematische Präzision' (mathematical precision), or 'mathematische Logik' (mathematical logic).
Seine Argumentation folgte einer klaren, fast mathematischen Logik.
His argument followed a clear, almost mathematical logic.
exact
'exakt' and 'mathematisch' can both have to do with great precision. 'exakt' is the general word when a detail, result, or piece of work is accurate and without deviation. 'mathematisch' first means: it belongs to mathematics or uses its methods. It can also be used figuratively when something seems very logical and precise, almost like a calculation. In normal everyday situations, 'exakt' is usually more natural. 'mathematisch' fits better when mathematics is really meant, or when you want to emphasize special rigor and logic.

Der Lehrer erklärt das Problem mit einer mathematischen Formel.
The teacher explains the problem with a mathematical formula.
Die Verkäuferin nennt den exakten Preis, bevor der Kunde bezahlt.
The saleswoman gives the exact price before the customer pays.