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The teacher asks us__listen to his explanations.
He need the students___write the composition.
The teacher wants us___use the dictionaries. The teacher let me__open my dictionary several times.
I make myself__write as neatly as I cold.
The teacher teacher makes us__work .
— или to

The teacher asks us to listen to his explanations.
Here, we use "to" after the verb "asks" because it is followed by the infinitive form of the verb "listen." The infinitive form of "listen" is "to listen."

He needs the students to write the composition.
Here, we use "to" after the verb "needs" because it is followed by the infinitive form of the verb "write." The infinitive form of "write" is "to write."

The teacher wants us to use the dictionaries.
Here, we use "to" after the verb "wants" because it is followed by the infinitive form of the verb "use." The infinitive form of "use" is "to use."

The teacher lets me open my dictionary several times.
Here, we use "to" after the verb "lets" because it is followed by the infinitive form of the verb "open." The infinitive form of "open" is "to open."

I make myself write as neatly as I could.
Here, we use "to" after the verb "make" because it is followed by the infinitive form of the verb "write." The infinitive form of "write" is "to write."

The teacher makes us work.
In this sentence, we do not need to use "to" because the verb "makes" is followed by the bare infinitive form of the verb "work."

In summary, the use of "to" after a verb depends on the specific verb used and its relationship to the following verb. Infinitive verbs in English are often used with the word "to" before them, but there are certain verbs that can be followed directly by the bare infinitive form without "to."