die Wurzel
root
The underground part of a plant that anchors it in the soil and absorbs water and nutrients.
Die Wurzeln dieses alten Baumes reichen sehr tief in die Erde.
The roots of this old tree reach very deep into the ground.

root
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root
The underground part of a plant that anchors it in the soil and absorbs water and nutrients.
Die Wurzeln dieses alten Baumes reichen sehr tief in die Erde.
The roots of this old tree reach very deep into the ground.

You can easily recognize the meaning of 'Wurzel' (root) from the context: words like 'Baum' (tree) or 'Erde' (soil/earth) point to the plant, 'Zahl' (number) to mathematics, and 'Problem' (problem) or 'Ursache' (cause) to the figurative sense.
Die Wurzel des Problems liegt oft in der Vergangenheit.
The root of the problem often lies in the past.
The adjective 'verwurzelt' (rooted) is derived from 'Wurzel' (root) and means that something is firmly anchored. It is often used figuratively for culture, traditions, or people.
Obwohl er lange im Ausland lebte, ist er tief in seiner Heimat verwurzelt.
Although he lived abroad for a long time, he is deeply rooted in his homeland.
The idiom 'Wurzeln schlagen' (to put down roots) means to settle in a new place and become at home there. It can be used both for plants and for people.
Nach einigen Jahren in der neuen Stadt hat sie dort endlich Wurzeln geschlagen.
After a few years in the new city, she finally put down roots there.
The idiom 'etwas an der Wurzel packen' (to tackle something at the root) means to fight the actual cause of a problem and not just the symptoms.
Anstatt nur die Folgen zu behandeln, müssen wir das Problem an der Wurzel packen.
Instead of only treating the consequences, we have to tackle the problem at the root.
In a mathematical context, 'Wurzel' (root) is almost always combined with the verb 'ziehen' (to draw/pull). You 'zieht die Wurzel aus' (take/find the root of) a number.
Im Kopf die dritte Wurzel aus 125 zu ziehen, ist nicht einfach.
Finding the cube root of 125 in your head is not easy.
reason / ground
'der Grund' and 'die Wurzel' can both explain why something happens or where something comes from. 'der Grund' is the normal word for a cause, occasion, or reason. You use it in many everyday situations. In this context, 'die Wurzel' refers more to the deep origin of a problem or development. It sounds more figurative and often a bit more fundamental. Learners may confuse the words because both ask about the why. In practice: for a direct explanation, you usually say 'der Grund'. If you mean the deepest cause, 'die Wurzel' fits better.

Die Wurzel des Streits liegt schon in einem alten Missverständnis.
The root of the argument already lies in an old misunderstanding.
Der Grund für die Verspätung war ein langer Stau auf der Autobahn.
The reason for the delay was a long traffic jam on the highway.