Haec exercitatio difficilis est, sed discipulis utilis est.

Haec means this here. It is a demonstrative adjective, and it modifies exercitatio.

So haec exercitatio = this exercise.

It is:

• nominative
• singular
• feminine

because it must match exercitatio in case, number, and gender.

Because exercitatio is a feminine noun.

In Latin, words like this, that, and adjectives in general must agree with the noun they describe. So since exercitatio is feminine singular, haec must also be feminine singular.

That is why you get:

• haec exercitatio = this exercise

not hic exercitatio or hoc exercitatio.

Exercitatio is nominative singular.

You can tell from its job in the sentence: it is the subject of est.

The basic structure is:

• Haec exercitatio difficilis est
• This exercise is difficult

So exercitatio is the thing being talked about, which makes it the subject, and subjects are in the nominative case.

Also, exercitatio is a third-declension noun.

Difficilis is an adjective, and it does agree with exercitatio in:

• gender
• number
• case

But not all Latin adjectives use different forms for masculine and feminine in the nominative singular.

Difficilis is a third-declension adjective of the two-termination type:

• masculine/feminine nominative singular: difficilis
• neuter nominative singular: difficile

So for a feminine noun like exercitatio, difficilis is exactly the correct form.

Because utilis often takes the dative in Latin.

Latin expresses useful to/for someone by putting that person in the dative:

• discipulis utilis = useful for the students / useful to the students

So discipulis is dative plural, meaning for the students.

This is a very common pattern:

• mihi utilis = useful to me
• nobis utilis = useful to us
• puero utilis = useful to the boy

No. Utilis agrees with exercitatio, not with discipulis.

The exercise is what is being described as useful. The students are not being described; they are the people for whom it is useful.

So:

• exercitatio = nominative singular feminine
• therefore utilis is nominative singular feminine

Meanwhile:

• discipulis is dative plural
• it depends on the meaning of utilis, not on adjective agreement

In other words:

• This exercise is useful
• useful for the students

Yes, Latin could leave it out in many contexts.

The full sentence is:

• Haec exercitatio difficilis est, sed discipulis utilis est.

Repeating est makes the structure clear and balanced:

• This exercise is difficult
• but it is useful for the students

Latin often does this.

But you may also see omission of the second est when it is easily understood:

• Haec exercitatio difficilis est, sed discipulis utilis.

That would still make sense. Repeating est is simply more explicit.

Because Latin has no articles like English the and a/an.

So exercitatio can mean:

• exercise
• an exercise
• the exercise

The exact sense depends on context.

In this sentence, haec already gives specificity, so haec exercitatio clearly means this exercise.

Latin word order is more flexible than English word order because Latin uses endings to show grammatical function.

So this sentence could be rearranged in different ways and still mean basically the same thing, for example:

• Haec exercitatio difficilis est, sed discipulis utilis est.
• Haec exercitatio est difficilis, sed discipulis est utilis.
• Difficilis est haec exercitatio, sed discipulis utilis est.

However, the original order is very natural and straightforward for a learner.

A useful way to read it is:

• Haec exercitatio = subject
• difficilis est = predicate
• sed = but
• discipulis utilis est = useful for the students

Sed is a conjunction meaning but.

It links the two clauses:

• Haec exercitatio difficilis est
• sed discipulis utilis est

So the sentence contrasts two ideas:

• the exercise is difficult
• but it is useful for the students

A helpful breakdown is:

• Haec — demonstrative adjective, nominative singular feminine, this
• exercitatio — noun, nominative singular feminine, exercise
• difficilis — adjective, nominative singular feminine, difficult
• est — verb, is
• sed — but
• discipulis — dative plural, for/to the students
• utilis — adjective, nominative singular feminine, useful
• est — is

So the structure is:

• This exercise is difficult
• but useful for the students

That sentence is a nice example of:

1. adjective agreement with the subject
2. the dative used with utilis
3. a simple contrast joined by sed