Magistra dicit prudentiam nec in vita publica tantum, sed etiam in vita privata utilem esse.

Because dicit here introduces an indirect statement. In Latin, indirect statement typically uses:

• an accusative subject
• plus an infinitive

So in Magistra dicit prudentiam ... utilem esse, the word prudentiam is the subject of esse, but in indirect statement that subject goes into the accusative, not the nominative.

If this were a direct statement, it would be:

• Prudentia utilis est. = Prudence is useful.

But after dicit, it becomes:

• Magistra dicit prudentiam utilem esse. = The teacher says that prudence is useful.

For the same reason: this is an indirect statement after dicit.

In indirect statement, Latin does not usually use that + finite verb the way English does. Instead, it uses:

• accusative subject + infinitive

So:

• direct: prudentia utilis est
• indirect: prudentiam utilem esse

Also notice that utilem agrees with prudentiam:

• prudentiam = feminine singular accusative
• utilem = feminine singular accusative

And esse is the infinitive to be.

The indirect statement is:

• prudentiam nec in vita publica tantum, sed etiam in vita privata utilem esse

That whole section depends on dicit.

So the sentence breaks down like this:

• Magistra dicit = The teacher says
• prudentiam ... utilem esse = that prudence is useful ...

Everything from prudentiam through esse is what the teacher says.

This is a balancing construction meaning:

• not only ... but also

In this sentence:

• nec in vita publica tantum
• sed etiam in vita privata

Together they mean:

• not only in public life, but also in private life

A learner may expect non ... sed etiam, and that would also be common in Latin in many contexts. Here nec gives the negative part of the contrast: not only in public life.

So the structure is:

• nec X tantum, sed etiam Y
• not only X, but also Y

Tantum means only or so much / only to such an extent, depending on context. Here it goes with the negative-and-additive pattern:

• nec ... tantum, sed etiam ...
• not only ..., but also ...

So in vita publica tantum on its own would mean something like only in public life, but with nec and sed etiam, the full sense becomes:

• not only in public life, but also in private life

Because they follow the preposition in, which here means in or within in a location/sphere sense.

• in vita publica = in public life
• in vita privata = in private life

When in indicates location, it normally takes the ablative.

So:

• vita publica = ablative singular
• vita privata = ablative singular

If in meant motion into something, it would usually take the accusative instead.

Because utilem agrees with prudentiam, not with the two in-phrases.

The thing being described as useful is prudentiam:

• prudence is useful
• useful in public life
• useful also in private life

The phrases in vita publica and in vita privata tell us where / in what sphere prudence is useful. They do not control the number of utilem.

So utilem is singular because prudentiam is singular.

Prudentia is a noun, first declension, feminine.

Its basic meaning is something like:

• prudence
• practical wisdom
• good judgment

In this sentence, it appears as prudentiam, the accusative singular form, because it is the subject accusative of the indirect statement.

Latin word order is more flexible than English word order. The infinitive esse often comes near the end of an indirect statement, especially after the predicate adjective or noun.

So this order is very natural:

• prudentiam ... utilem esse

Latin often likes to place the key finite verb first or early and then let the dependent clause unfold, with esse closing the indirect statement.

In some Latin expressions, forms of to be can be omitted, especially in poetry or very compressed styles. But in a standard prose sentence like this, esse is normally expressed in indirect statement.

So here utilem esse is the expected full form.

Yes. Both are adjectives agreeing with vita.

• vita publica = public life
• vita privata = private life

Agreement is in:

• gender: feminine
• number: singular
• case: ablative

So both adjective-noun pairs are grammatically matched.

Yes, and that is often a very helpful strategy.

The basic statement is:

• Prudentia nec in vita publica tantum, sed etiam in vita privata utilis est.
• Prudence is useful not only in public life, but also in private life.

Then after Magistra dicit, Latin changes it into indirect statement:

• prudentia → prudentiam
• utilis est → utilem esse

So:

• Magistra dicit prudentiam nec in vita publica tantum, sed etiam in vita privata utilem esse.

That is a very useful transformation to practice.