Ich weise den Zählerstand mit einem Foto nach.

Because nachweisen is a separable verb.

Its full dictionary form is nachweisen. In a normal main clause, German splits separable verbs into:

• the conjugated base verb: weise
• the prefix: nach

So:

• Ich weise den Zählerstand mit einem Foto nach.

Compare:

• Infinitive: Ich möchte den Zählerstand mit einem Foto nachweisen.
• Main clause: Ich weise den Zählerstand mit einem Foto nach.

This is very common in German with verbs like anrufen, aufstehen, einkaufen, and nachweisen.

Nachweisen means something like to prove, to demonstrate, to provide evidence of, or to verify with evidence.

In this sentence, it has the sense of documenting or proving something by means of supporting evidence. A photo is the evidence.

So nachweisen is a bit more formal and specific than a simple verb like zeigen (to show). It suggests that you are not just showing something casually; you are providing proof.

Because den Zählerstand is the direct object of the verb, so it is in the accusative case.

The noun is:

• nominative: der Zählerstand
• accusative: den Zählerstand

Here, the thing being proven/documented is the meter reading, so German uses the accusative:

• Ich weise den Zählerstand nach.

If it were the subject, it would be der Zählerstand.

Because the preposition mit always takes the dative case.

The noun das Foto is neuter, so:

• nominative: ein Foto
• dative: einem Foto

That is why the sentence says:

• mit einem Foto

This phrase tells you the means or instrument: with a photo.

Zählerstand is a compound noun:

• Zähler = meter, counter
• Stand = level, reading, status, position

So literally it is something like meter status/reading, and in normal English it is meter reading.

German often forms nouns by combining smaller words like this. Once you get used to compounds, they become much easier to understand.

It is the correct 1st person singular present tense form of weisen:

• ich weise
• du weist
• er/sie/es weist
• wir weisen
• ihr weist
• sie/Sie weisen

With the separable prefix, you get:

• ich weise ... nach

So the sentence uses a normal present-tense form, but with a separable verb.

The basic structure is:

• Ich = subject
• weise = conjugated verb
• den Zählerstand = direct object
• mit einem Foto = prepositional phrase
• nach = separable prefix

So the pattern is:

Subject + conjugated verb + object + additional information + separable prefix

That is a very typical German main-clause pattern.

You could also move some elements for emphasis, for example:

• Mit einem Foto weise ich den Zählerstand nach.

But in a standard neutral sentence, the original order is very natural.

Yes, but only in structures where German normally uses an infinitive or puts the verb at the end.

For example:

• Ich möchte den Zählerstand mit einem Foto nachweisen.
• ..., weil ich den Zählerstand mit einem Foto nachweise.

In the first example, nachweisen stays together because it is an infinitive. In the second example, the verb is in a subordinate clause, so the full verb goes to the end as nachweise.

So whether it splits depends on the grammar of the sentence.

Yes, it sounds fairly formal or administrative.

In everyday speech, people might sometimes use other verbs depending on the situation, such as:

• zeigen = to show
• belegen = to document / substantiate
• fotografieren = to photograph

But nachweisen is very natural in official, technical, or utility-related contexts, especially when you are proving something with evidence.

So in a meter-reading context, it sounds quite appropriate.

German does not need any extra linking word there. The phrase mit einem Foto simply attaches directly to the verb phrase and tells you how the proof is provided.

So German says, very naturally:

• Ich weise den Zählerstand mit einem Foto nach.

English and German often build these instrumental phrases in similar ways:

• with a photo
• mit einem Foto

No extra word is needed.

The perfect tense of nachweisen uses haben and the past participle nachgewiesen:

• Ich habe den Zählerstand mit einem Foto nachgewiesen.

Notice that in the participle, the prefix is no longer separated. It becomes one word again:

• infinitive: nachweisen
• present main clause: weise ... nach
• past participle: nachgewiesen

That pattern is very useful to learn for separable verbs.