Eine Wurzel kommt unter dem Busch aus dem Boden.

Because Wurzel is a feminine noun in German: die Wurzel.

So in the nominative singular, the indefinite article is:

• eine Wurzel = a root

Compare:

• ein Busch = a bush (masculine)
• eine Wurzel = a root (feminine)
• ein Blatt = a leaf (neuter)

In this sentence, Eine Wurzel is the subject, so it is in the nominative case.

Because unter is a two-way preposition. That means it can take:

• dative for location
• accusative for direction/movement toward a place

Here, unter dem Busch describes where the root is appearing: under the bush. It is a location, so German uses the dative:

• unter dem Busch = under the bush

If it were about movement to a position under the bush, you would use the accusative:

• Die Katze läuft unter den Busch. = The cat runs under the bush.

So even though the root is coming out, unter dem Busch still tells you the location where this happens, not the destination of the movement.

Because aus always takes the dative case.

So:

• aus dem Boden = out of the ground
• aus dem Haus = out of the house
• aus der Tasche = out of the bag

Here, dem Boden is dative because of aus.

It means comes out of the ground or emerges from the ground.

The core idea is:

• kommen = to come
• aus = out of / from
• aus dem Boden kommen = to come out of the ground

So the sentence describes a root that is visible because it is emerging from the earth.

No. In this sentence, the verb is just kommen.

German also has the verb auskommen, but that usually means things like:

• to get by
• to manage
• to come along with

That is not what is happening here.

In Eine Wurzel kommt unter dem Busch aus dem Boden, aus dem Boden is a normal prepositional phrase meaning out of the ground.

German word order is flexible, especially after the verb. Both phrases are giving extra information:

• unter dem Busch = where
• aus dem Boden = from where / out of what

The sentence as given sounds natural and descriptive:

• Eine Wurzel kommt unter dem Busch aus dem Boden.

It first tells you the general location (under the bush) and then the source of emergence (out of the ground).

You could sometimes rearrange parts in German, but the original order is perfectly normal.

Because German main clauses normally follow the V2 rule: the finite verb goes in the second position.

Here the first element is:

• Eine Wurzel

So the verb comes next:

• kommt

Then the remaining information follows:

• unter dem Busch aus dem Boden

This is a basic German pattern:

• Der Hund schläft im Garten.
• Ein Vogel sitzt auf dem Baum.

Yes, it means the root is appearing beneath / under the bush.

In natural English, under the bush may sound slightly odd if you imagine the root belongs to the bush itself, but in German the phrase simply identifies the place where the root is visible.

So the image is something like: beneath the bush, a root is sticking out or emerging from the soil.

Both can relate to ground/earth/soil, but they are not always identical.

• Boden often means ground, floor, or surface
• Erde often means earth, soil, or even the Earth

In this sentence, aus dem Boden is a very natural expression for out of the ground.

You could hear aus der Erde too in some contexts, but aus dem Boden is especially common when describing something emerging from the ground surface.

The basic form is:

• der Busch = the bush

It is masculine.

That is why:

• nominative: der Busch
• accusative: den Busch
• dative: dem Busch

In this sentence, it becomes dem Busch because unter takes the dative here.

Yes. Wurzel can mean:

• a root of a plant
• a root in mathematics, as in square root
• more figuratively, a source or origin

In this sentence, the meaning is clearly the plant sense because of Busch and Boden.

Yes, and in many contexts that may even sound a bit more natural.

• kommen aus dem Boden = to come out of the ground
• wachsen aus dem Boden = to grow out of the ground

Kommt aus dem Boden focuses on the root appearing / emerging. Wächst aus dem Boden focuses more on the process of growing.

Both are understandable, but they are not exactly identical in emphasis.

It is indefinite, because it uses eine:

• Eine Wurzel = a root

It does not mean:

• Die Wurzel = the root

So the sentence introduces a root that has not been specifically identified before.

Yes, but only if the phrase expresses movement to a position under something.

Compare:

• Die Wurzel liegt unter dem Busch.
  The root is lying under the bush.
  → location → dative

• Das Tier kriecht unter den Busch.
  The animal crawls under the bush.
  → movement toward that position → accusative

In your sentence, the movement is not toward the bush. The root is emerging from the ground, and under the bush is just the location, so dative is used.

Very literally, it breaks down like this:

• Eine Wurzel = a root
• kommt = comes
• unter dem Busch = under the bush
• aus dem Boden = out of the ground

So a very literal English rendering would be:

• A root comes out of the ground under the bush.

More natural English might be:

• A root is coming out of the ground under the bush.
• A root sticks out of the ground under the bush.

Depending on context, English often chooses a slightly different verb to sound more natural.