אם שואב האבק לא עובד, אני אנקה את הרצפה עם המטאטא ועם מטלית.

This is a very common Hebrew structure called a construct chain (סמיכות).

• שואב = literally sucker / sucking
• אבק = dust

So שואב אבק literally means dust-sucker, which is the normal Hebrew way to say vacuum cleaner.

When a construct chain is definite, Hebrew usually puts ה־ only on the second noun:

• שואב אבק = a vacuum cleaner
• שואב האבק = the vacuum cleaner

So שואב stays without ה־, and האבק makes the whole phrase definite.

A similar pattern:

• בית ספר = a school
• בית הספר = the school

Because לא עובד here describes the vacuum cleaner’s state: it isn’t working / doesn’t work.

In Hebrew, the present tense often covers both:

• simple present: doesn’t work
• progressive meaning: isn’t working

So אם שואב האבק לא עובד sounds very natural for if the vacuum cleaner isn’t working.

If you said לא יעבוד, that would mean more like won’t work in the future.

So the sentence is basically:

• If the vacuum cleaner isn’t working, I’ll clean the floor...

Yes. עובד is very commonly used for both:

• a person who works
• a machine/device that works

So:

• אני עובד = I work / I am working
• המחשב עובד = the computer works / is working
• שואב האבק לא עובד = the vacuum cleaner doesn’t work / isn’t working

That is very natural Hebrew.

You may also hear פועל for machines:

• שואב האבק לא פועל

That also means the vacuum cleaner isn’t operating, and can sound a bit more technical or formal.
In everyday speech, לא עובד is extremely common.

את is the direct object marker. It appears before a definite direct object.

Here, הרצפה means the floor, and it is definite because of ה־, so Hebrew uses את:

• אני אנקה את הרצפה = I will clean the floor

Important points:

• את does not mean with
• It usually does not get translated into English
• It is used when the direct object is definite

Compare:

• אני אנקה רצפה = I will clean a floor
• אני אנקה את הרצפה = I will clean the floor

אנקה is the future tense, first person singular: I will clean.

It comes from the verb לנקות = to clean.

So:

• אני אנקה = I will clean
• אתה תנקה = you will clean
• הוא ינקה = he will clean

A useful point for learners: with אני, the future form is the same whether the speaker is male or female.
So both a man and a woman can say:

• אני אנקה

Hebrew often repeats prepositions before each item in a list, especially in careful or clear speech.

So this is natural:

• עם המטאטא ועם מטלית

It makes it very clear that both items are tools being used.

You can also sometimes hear a shorter version:

• עם המטאטא ומטלית

That is also understandable, but repeating עם is very normal and often sounds smoother or clearer.

Because the sentence is treating them differently:

• המטאטא = the broom
• מטלית = a cloth

So the speaker seems to mean something like:

• with the broom and with a cloth

That is perfectly possible in Hebrew. Maybe the broom is a specific, known broom, while the cloth is just any cloth.

If you wanted both to be definite, you could say:

• עם המטאטא ועם המטלית = with the broom and the cloth

If you wanted both to be indefinite, you could say:

• עם מטאטא ועם מטלית = with a broom and a cloth

מטלית means a cloth, often a cleaning cloth, rag, or wipe depending on context.

In this sentence, it suggests something used for cleaning the floor along with the broom.

It is not the same as:

• מטאטא = broom
• מגב = squeegee / floor wiper
• מופ = mop
• סמרטוט = rag

So מטלית is a fairly general word for a cloth used for cleaning.

Yes. Hebrew allows that.

The original order:

• אם שואב האבק לא עובד, אני אנקה את הרצפה עם המטאטא ועם מטלית.

You could also say:

• אני אנקה את הרצפה עם המטאטא ועם מטלית אם שואב האבק לא עובד.

Both are grammatical.

The version with אם first is often a little more natural when you want to present the condition before the result:

• If the vacuum cleaner isn’t working, I’ll clean the floor...

Yes. Hebrew often drops subject pronouns when the verb already makes the subject clear.

So both are possible:

• אני אנקה את הרצפה = I will clean the floor
• אנקה את הרצפה = I will clean the floor

Because אנקה already tells you the subject is I, the pronoun אני is optional.

Keeping אני can add a bit of emphasis or clarity:

• I’ll clean the floor

Leaving it out often sounds a little more natural in ordinary speech.