Learn Hardcore Russian: Я точно знаю перевод этого слова. - I know the exact translation of this word.

Because in Russian we express “the translation of something” by putting the “something” into the genitive. Here перевод (translation) is the head noun and этого слова means “of this word,” so слова takes the genitive.

What does точно mean in this sentence? Is it “exactly” or “certainly”?

Here точно means “certainly” or “for sure,” emphasizing the speaker’s confidence. It doesn’t mean “precisely” in the sense of tiny details, but rather “I am absolutely sure.”

Could I say Я знаю точно перевод этого слова instead of Я точно знаю перевод этого слова?

Technically yes, Russian word order is flexible, but Я точно знаю… is more natural. Placing точно right after я highlights your certainty in the most idiomatic way.

What’s the difference between перевод and значение? Can I use значение here?

Перевод is “translation” (the equivalent in another language). Значение is “meaning” (the concept or definition in the same language). If you want to say “I know the definition,” you’d use значение:
Я точно знаю значение этого слова.
But for “I know the translation,” you need перевод.

Why don’t we use any preposition before перевод?

In Russian, the verb знать takes a direct object without a preposition. You simply say знать перевод, not знать о переводе or знать для перевода.

Could I drop этого and say Я точно знаю перевод слова?

Yes, Я точно знаю перевод слова is grammatically correct and means the same, but omitting этого makes it less specific. этого слова points to a particular word you’re discussing.

Why is the noun перевод used instead of the verb переводить?

Because you’re stating “I know the translation” (noun). If you wanted to say “I can translate this word,” you would use the verb:
Я могу перевести это слово.
Here, however, you’re focusing on knowing the translation itself.