Hoc exercitium difficilius est quam illud.

Because exercitium is a neuter noun.

The demonstrative hic, haec, hoc changes form to match the noun it goes with in gender, number, and case. Since exercitium is:

• neuter
• singular
• nominative (the subject of the sentence)

the correct form is hoc.

So:

• hic = masculine nominative singular
• haec = feminine nominative singular
• hoc = neuter nominative singular

Because exercitium is neuter, Latin uses hoc exercitium = this exercise.

Exercitium is nominative singular.

You can tell from its role in the sentence: it is the thing being described as more difficult. In English, it is the subject: this exercise.

The verb is est = is, so the basic structure is:

• Hoc exercitium = subject
• difficilius = predicate adjective
• est = verb
• quam illud = comparison phrase

Also, exercitium is a second-declension neuter noun, and its nominative singular ends in -um.

Because difficilius is the neuter nominative singular form of the comparative adjective.

The comparative of difficilis = difficult is:

• masculine/feminine: difficilior
• neuter: difficilius

Since it describes exercitium, which is neuter singular, the adjective must also be neuter singular. So:

• exercitium difficilius = a more difficult exercise

This is just agreement: the adjective matches the noun it describes.

Literally, difficilius means more difficult.

It is the comparative form of difficilis:

• difficilis = difficult
• difficilior / difficilius = more difficult

So the sentence is built around a comparison:

• Hoc exercitium difficilius est = This exercise is more difficult
• quam illud = than that one

Latin, like English, often leaves out a noun when it is obvious from context.

So illud here really means:

• illud exercitium = that exercise

But Latin does not need to repeat exercitium, because the comparison already makes it clear.

English does the same thing:

• This exercise is harder than that one

So illud by itself means that one or more specifically that exercise.

Because illud is standing for that exercise, and it is being compared directly with hoc exercitium.

In this sentence, quam means than, and it introduces the second item in the comparison:

• Hoc exercitium = this exercise
• quam illud = than that one / than that exercise

Since illud refers to exercitium, it is also neuter singular.

A learner may expect a different case after than, because English sometimes changes pronouns in formal grammar discussions, but here Latin simply uses illud to match the omitted noun exercitium.

Not always, but in this sentence it does.

Quam has several uses in Latin, but one very common use is with comparatives:

• difficilius ... quam ... = more difficult than ...
• maior quam ... = bigger than ...
• celerior quam ... = faster than ...

So here quam is the standard comparison word meaning than.

Yes, Latin can also make comparisons by using the ablative of comparison instead of quam.

For example, Latin could say:

• Hoc exercitium illo difficilius est.

That means the same thing: This exercise is more difficult than that one.

Here:

• illo is ablative
• there is no quam

Both patterns are common:

• difficilius quam illud
• illo difficilius

For many learners, the quam version is easier to recognize at first because it looks more like English.

Yes. Latin word order is much more flexible than English word order because the endings show the grammatical relationships.

The sentence as given is:

• Hoc exercitium difficilius est quam illud.

That is a very natural and clear order. But Latin could also rearrange it, for example:

• Difficilius est hoc exercitium quam illud.
• Hoc difficilius est exercitium quam illud.
  (less basic, but possible in the right context)

The most important thing is not the position of the words, but their forms:

• hoc goes with exercitium
• difficilius agrees with exercitium
• illud refers to the omitted exercitium

So the original order is standard and readable, but not the only possible one.

In a normal teaching sentence like this, yes, est is expected.

Latin does sometimes leave out forms of to be, especially in poetry, very compressed style, or where the meaning is obvious. But in straightforward prose, est is normally included.

So:

• Hoc exercitium difficilius est quam illud = normal full sentence

Leaving out est would be much less standard for a beginner sentence.

Here are the basic dictionary forms:

• hoc → from hic, haec, hoc = this
• exercitium → exercitium, exercitii n. = exercise
• difficilius → comparative of difficilis, difficile = difficult
• est → from sum, esse = to be
• quam = than / as
• illud → from ille, illa, illud = that

A very useful habit in Latin is to identify which words are inflected forms and which are the dictionary entries behind them.

A helpful way to break it up is:

• Hoc exercitium = this exercise
• difficilius est = is more difficult
• quam illud = than that one

So the grammar pattern is:

this + noun + comparative adjective + is + than + that one

This is a very common Latin comparison pattern, and once you recognize it, other sentences built the same way become much easier to read.