Mi tranĉas tomaton per akra tranĉilo.

The ending -n marks the direct object in Esperanto.

In this sentence, tomaton is the thing being cut, so it gets -n:

• Mi = I
• tranĉas = cut / am cutting
• tomaton = a tomato / the tomato, as the object

This is called the accusative ending. English usually shows the object by word order, but Esperanto often shows it with -n.

Because akra tranĉilo is not the direct object here. It is part of a per phrase:

• per akra tranĉilo = with a sharp knife / by means of a sharp knife

The knife is the instrument used to do the action, not the thing being cut. So it does not take the accusative ending.

If the knife were the direct object, then it would take -n. For example:

• Mi prenas akran tranĉilon. = I take a sharp knife.

There, tranĉilon is the direct object, so both the noun and its adjective get -n:

• akran tranĉilon

Per means by means of, using, or often simply with when talking about the tool or method used.

So:

• per akra tranĉilo = using a sharp knife

This is different from kun, which usually means with in the sense of together with someone or something.

Compare:

• Mi tranĉas tomaton per tranĉilo. = I cut a tomato with a knife.
• Mi manĝas kun amiko. = I eat with a friend.

A common learner mistake is to use kun for tools. For instruments, per is usually the right choice.

In Esperanto, all present-tense verbs end in -as.

So:

• tranĉi = to cut
• tranĉas = cut / am cutting / cuts / is cutting

The verb ending does not change for different persons:

• mi tranĉas = I cut
• vi tranĉas = you cut
• li tranĉas = he cuts
• ili tranĉas = they cut

This is much simpler than English, because the verb form stays the same.

Esperanto has la for the, but it has no separate word for a/an.

So:

• tomato can mean a tomato or just tomato in a general sense
• la tomato means the tomato

That means:

• Mi tranĉas tomaton = I am cutting a tomato / I cut tomato
• Mi tranĉas la tomaton = I am cutting the tomato

If the meaning has already been shown, the sentence without la is completely normal.

Yes. That is correct.

Adding la makes the tomato specific:

• Mi tranĉas tomaton = I’m cutting a tomato
• Mi tranĉas la tomaton = I’m cutting the tomato

You could also say:

• Mi tranĉas la tomaton per la akra tranĉilo

but that sounds more specific, as if both the tomato and the knife are known in the situation.

Yes. In Esperanto, adjectives can come before or after the noun.

So both are possible:

• akra tranĉilo
• tranĉilo akra

But putting the adjective before the noun is more common and usually sounds more natural in simple sentences.

Also remember that adjectives must agree with the noun in number and case. Here both are singular and neither has -n, so:

• akra tranĉilo

If the noun had -n, the adjective would too:

• Mi prenas akran tranĉilon.

To make a noun plural in Esperanto, add -j.

So:

• tomato = tomato
• tomatoj = tomatoes

If the plural noun is also a direct object, it takes both -j and -n:

• tomatojn = tomatoes, as a direct object

Example:

• Mi tranĉas tomatojn. = I cut tomatoes / I am cutting tomatoes.

If there is an adjective, it must match:

• Mi tranĉas ruĝajn tomatojn. = I cut red tomatoes.

Yes, Esperanto word order is fairly flexible because the -n ending helps show what the object is.

The most neutral order is:

• Mi tranĉas tomaton per akra tranĉilo.

But other orders are possible, for example:

• Tomaton mi tranĉas per akra tranĉilo.
• Per akra tranĉilo mi tranĉas tomaton.

These can change the emphasis, but the basic meaning stays clear because tomaton still has -n.

That said, beginners should usually stick to the normal order: subject + verb + object + other phrases.

A few pronunciation points matter here:

• ĉ is pronounced like ch in church
• c by itself is pronounced like ts in cats
• stress in Esperanto is always on the second-to-last syllable

So:

• tranĉas is roughly tran-CHAS
• tranĉilo is roughly tran-CHEE-lo, with stress on ĉi: tran-CHI-lo

Also:

• akra is roughly AH-kra
• tomaton is roughly to-ma-TOHN, with stress on ma

The dictionary forms are:

• mi = I
• tranĉi = to cut
• tomato = tomato
• per = by means of, using
• akra = sharp
• tranĉilo = knife

A useful thing to notice is that tranĉilo is built from parts:

• tranĉ- = cut
• -il- = tool or instrument
• -o = noun ending

So tranĉilo literally means cutting tool. This kind of word-building is very common in Esperanto and helps a lot with vocabulary learning.