Learn Hardcore Greek: Αυτές οι ασκήσεις είναι καλό παράδειγμα για το πώς μπορούμε να θυμόμαστε καλύτερα τις νέες λέξεις. - These exercises are a good example of how we can remember the new words better.

You wouldn’t normally say Αυτές ασκήσεις on its own in standard modern Greek; it sounds incomplete or incorrect. The article οι is needed with a common noun like ασκήσεις.

You can, however, move the demonstrative after the noun:

Οι ασκήσεις αυτές = these exercises (same meaning, slightly different emphasis)

So the main natural options are:

Αυτές οι ασκήσεις
Οι ασκήσεις αυτές

but not Αυτές ασκήσεις.

Why is it Αυτές and not Αυτοί or Αυτά?

The choice of demonstrative depends on the gender and number of the noun it refers to.

άσκηση is feminine:
- singular: η άσκηση
- plural: οι ασκήσεις

So:

αυτή η άσκηση (this exercise)
αυτές οι ασκήσεις (these exercises)

Compare with other genders:

Masculine plural: αυτοί οι μαθητές (these students)
Neuter plural: αυτά τα βιβλία (these books)

Because ασκήσεις is feminine plural, the correct demonstrative is Αυτές.

Why does the sentence say είναι καλό παράδειγμα in the singular, even though ασκήσεις is plural?

The subject is plural (Αυτές οι ασκήσεις), but the complement after είναι is a single example:

καλό παράδειγμα = a good example (one example)

Greek here is saying:

These exercises are (a) good example (singular)

You are treating the whole group of exercises as one example of a method.

If you wanted to talk about multiple examples, you would use the plural:

είναι καλά παραδείγματα = they are good examples

But that would change the meaning (several separate examples), which is not what this sentence is doing. It’s emphasizing the set of exercises as one model/example.

Why is there no ένα (as in είναι ένα καλό παράδειγμα)?

Both are grammatically correct:

είναι καλό παράδειγμα
είναι ένα καλό παράδειγμα

In Greek, the indefinite article (ένα, ένας, μια) is optional, especially in predicate positions after είμαι (to be). Leaving it out is very natural and common:

Είναι δάσκαλος. = He is a teacher.
Είναι καλό παράδειγμα. = It is a good example.

Adding ένα can make it a bit more specific or emphatic:

Είναι ένα καλό παράδειγμα (it is a good example, one of the good examples)

In everyday speech, είναι καλό παράδειγμα is perfectly natural.

Why is the preposition για used with παράδειγμα here?

The phrase παράδειγμα για κάτι is very common:

παράδειγμα για το πώς… = an example of how… / an example for how…

για often corresponds to English for or of in such structures:

παράδειγμα για μίμηση = an example to imitate
παράδειγμα για το τι πρέπει να κάνουμε = an example of what we must do

So καλό παράδειγμα για το πώς μπορούμε… literally means:

a good example for how we can…

You could also see παράδειγμα του πώς… in some contexts, but για το πώς is very natural and common.

What is the function of το in για το πώς? Why not just για πώς?

The το here is the neuter definite article, and it does something important: it turns the whole clause into a noun-like phrase.

πώς on its own is an interrogative adverb (how?).
το πώς means something like the way (that) / the manner in which / how in a noun-like way.

With για:

για το πώς μπορούμε να θυμόμαστε…
≈ about/how we can remember…
literally: for the how we can remember…

Greek often uses το to nominalize a subordinate clause:

το τι είπε = the (fact of) what he said
το πότε θα φύγουμε = the (question of) when we will leave

So για πώς would be ungrammatical here; you need για το πώς.

What exactly is να θυμόμαστε? Is it a subjunctive, and why is να needed?

Να θυμόμαστε is the present subjunctive (or “subjunctive-like” form) of the verb θυμάμαι (to remember).

Greek does not have an infinitive like English to remember; instead it uses να + verb:

να θυμόμαστε ≈ (to) remember / (in order that) we remember / so that we keep remembering

Here, μπορούμε να θυμόμαστε literally means we can be remembering / we are able to remember (habitually or better in general).

A few points:

θυμάμαι is a so‑called deponent verb: it looks middle/passive but is active in meaning.
The present subjunctive (να θυμόμαστε) suggests an ongoing or repeated process (remembering in general, improving memory overall).
The aorist subjunctive would be να θυμηθούμε, which focuses on one act of remembering (to manage to remember once, to recall at a particular time).

If the sentence said:

…για το πώς μπορούμε να θυμηθούμε τις νέες λέξεις.

the nuance would be more like how we can manage to remember the new words (on a specific occasion). Using να θυμόμαστε fits better with the idea of remembering them better in general, as a skill or habit.

Could we also say Εμείς μπορούμε να θυμόμαστε…? Why is εμείς omitted?

Yes, grammatically you can say:

Εμείς μπορούμε να θυμόμαστε καλύτερα τις νέες λέξεις.

But in normal Greek, subject pronouns (εγώ, εσύ, εμείς etc.) are often dropped, because the verb ending already shows the person and number:

μπορούμε clearly indicates we.

You only normally include εμείς if you want to emphasize the subject:

Εμείς μπορούμε να θυμόμαστε…
= We can remember… (in contrast to others, or stressing that it’s us)

In this sentence, no such contrast is needed, so μπορούμε without εμείς is more natural.

Can we move καλύτερα in the sentence? For example: να θυμόμαστε τις νέες λέξεις καλύτερα?

Yes. Adverbs like καλύτερα have fairly flexible position in Greek. These are all acceptable:

να θυμόμαστε καλύτερα τις νέες λέξεις
να θυμόμαστε τις νέες λέξεις καλύτερα

The version in the original sentence:

να θυμόμαστε καλύτερα τις νέες λέξεις

puts καλύτερα right after the verb, which is very natural and common.

Small nuance:

να θυμόμαστε καλύτερα τις νέες λέξεις slightly focuses on remembering better as an ability.
να θυμόμαστε τις νέες λέξεις καλύτερα may sound a bit more like these specific words, better than before, but in most contexts the difference is minimal.

Both word orders are fine.

Why is it τις νέες λέξεις and not another form? What is going on with the case and agreement?

Τις νέες λέξεις is in the accusative plural feminine, because it is the direct object of να θυμόμαστε (to remember):

Verb: θυμόμαστε (we remember)
Object: τις νέες λέξεις (the new words)

Breakdown:

λέξεις is the plural of η λέξη (word).
- nominative plural: οι λέξεις
- accusative plural: τις λέξεις
νέες is the feminine plural form of the adjective νέος, νέα, νέο (new).
In the accusative plural feminine, it is also νέες.
The article τις must match the noun in gender, number and case:
- feminine, plural, accusative → τις

So we have full agreement:

τις (article, fem pl acc) + νέες (adj, fem pl acc) + λέξεις (noun, fem pl acc)

What is the difference between νέες λέξεις and καινούριες λέξεις? Do they both mean “new words”?

Both are commonly used and can often be translated as new words, but there is a nuance:

νέος, νέα, νέο
- new in time, recent, young
- in this context: newly learned/recently introduced words
καινούριος, καινούρια, καινούριο
- brand new, unused, new compared to something old

In practice, with λέξεις, both can appear:

νέες λέξεις: stresses that these are new (additional) words you are learning or have just encountered.
καινούριες λέξεις: can also mean new words, often with a flavor of brand‑new / you didn’t know them before.

In language‑learning contexts, νέες λέξεις is extremely normal and perhaps a bit more neutral/formal, which suits a sentence like this well.

Could the sentence also be Αυτές οι ασκήσεις είναι ένα καλό παράδειγμα για το πώς μπορούμε να θυμόμαστε καλύτερα τις νέες λέξεις? Would that change the meaning?

Yes, that version is fully correct:

Αυτές οι ασκήσεις είναι ένα καλό παράδειγμα για το πώς μπορούμε να θυμόμαστε καλύτερα τις νέες λέξεις.

Adding ένα does not change the basic meaning; it still means:

These exercises are a good example of how we can remember new words better.

The difference is stylistic:

Without ένα: slightly more general, very natural, a bit more compact.
With ένα: marginally more specific/emphatic (they are a good example).

Both forms would be understood the same way in most real situations.