W zeszycie zaznaczam nowe słowa, żeby łatwiej je zapamiętać.

Because the preposition w meaning in normally requires the locative case when it shows location.

• dictionary form: zeszyt = notebook
• locative singular: zeszycie

So:

• w zeszycie = in the notebook

This is a very common pattern:

• w domu = in the house
• w szkole = at school / in school
• w zeszycie = in the notebook

Yes, w is flexible and often translates as in, at, or sometimes even inside, depending on context.

Here, w zeszycie most naturally means:

• in the notebook
• sometimes idiomatically in my notebook

In English, we might also say in a notebook rather than on a notebook, because the writing is inside it, on its pages.

Zaznaczam is the 1st person singular present tense of zaznaczać.

It can mean:

• I mark
• I highlight
• I indicate
• I underline / pick out (depending on context)

In this sentence, it means something like:

• I mark new words
• I highlight new words

So the speaker is doing something in the notebook to make the words stand out.

In Polish, the present tense is often used for:

• something you do regularly
• a habit
• a general practice

So W zeszycie zaznaczam nowe słowa means something like:

• I mark new words in my notebook
• I usually mark new words in my notebook
• I mark new words in the notebook

It does not have to mean only right now at this moment. It can describe a routine.

It is the object of the verb, so it is in the accusative case. But in this particular phrase, the accusative looks the same as the nominative.

• nominative: nowe słowa = new words
• accusative: nowe słowa = new words

This happens because:

• słowa is the plural of słowo
• for non-masculine-personal plural nouns, the nominative and accusative are often identical

So although the form does not change visibly, grammatically it is functioning as the accusative object.

Żeby introduces a purpose clause. It often means:

• so that
• in order to
• so as to

So:

• żeby łatwiej je zapamiętać = so that it is easier to remember them
• more naturally in English: to remember them more easily

It connects the action with its purpose:

• I mark new words ... so that I can remember them more easily.

Because łatwiej is an adverb, while łatwiejszy is an adjective.

Here we need an adverb because it describes how the remembering happens:

• łatwo = easily
• łatwiej = more easily

Compare:

• To jest łatwiejsze zadanie. = This is an easier task.
  
  • adjective: łatwiejsze
• Chcę to łatwiej zapamiętać. = I want to remember it more easily.
  
  • adverb: łatwiej

So in your sentence, łatwiej is correct because it modifies the action zapamiętać.

Je means them. It refers back to nowe słowa.

• nowe słowa = new words
• je = them

So:

• żeby łatwiej je zapamiętać = to remember them more easily

Grammatically, je is the accusative plural pronoun here.

In Polish, object pronouns like go, ją, je, ich often come before the infinitive or before the main verb. This is very natural word order.

So:

• łatwiej je zapamiętać = natural
• literally: more easily them remember

English places them after the verb, but Polish often places the pronoun earlier.

That said, Polish word order is flexible, and different order can be used for emphasis. But łatwiej je zapamiętać is the most neutral and natural version here.

This is an aspect question.

• pamiętać = to remember, to have something in memory
• zapamiętać = to memorize, to learn and retain, to remember successfully

In this sentence, the speaker means the goal of getting the words into memory, so zapamiętać fits better.

Compare:

• Pamiętam to słowo. = I remember this word.
• Chcę zapamiętać to słowo. = I want to memorize / remember this word.

After żeby in a purpose clause, Polish very often uses the perfective infinitive when the goal is a completed result. Here, the desired result is: the new words become remembered.

Polish often omits subject pronouns when they are clear from the verb ending.

• zaznaczam already means I mark
• the ending -am tells you the subject is I

So ja is unnecessary unless you want emphasis or contrast.

Compare:

• Zaznaczam nowe słowa. = I mark new words.
• Ja zaznaczam nowe słowa, a on je przepisuje. = I mark new words, and he copies them out.

In the second sentence, ja is added for contrast.

Yes. Polish word order is fairly flexible, although some versions sound more natural than others.

Your sentence:

• W zeszycie zaznaczam nowe słowa, żeby łatwiej je zapamiętać.

Other possible versions:

• Zaznaczam nowe słowa w zeszycie, żeby łatwiej je zapamiętać.
• Nowe słowa zaznaczam w zeszycie, żeby łatwiej je zapamiętać.

The differences are mostly about focus and emphasis:

• W zeszycie first: emphasizes where
• Zaznaczam first: more neutral action-first statement
• Nowe słowa first: emphasizes what is being marked

The original sentence is perfectly natural.

Because żeby introduces a subordinate clause, and in Polish such clauses are normally separated by a comma.

So the structure is:

• main clause: W zeszycie zaznaczam nowe słowa
• subordinate purpose clause: żeby łatwiej je zapamiętać

That is why the comma is required.

Yes. The subject is understood from context.

The full idea is something like:

• żeby [ja mógł / mogła] łatwiej je zapamiętać
• so that I can remember them more easily

But Polish often leaves this unstated when it is obvious. The infinitive zapamiętać allows the sentence to stay compact and natural.

So even though the subject is not explicitly repeated, we understand that the same person who marks the words is the one who wants to remember them.