Мне нужен точный перевод этого слова.

Because Russian uses a different structure from English here.

Мне нужен X literally means something like To me, X is needed.

So:

• мне = to me / for me (dative case)
• нужен = needed
• точный перевод этого слова = an exact translation of this word

A native English speaker often expects I need..., but Russian commonly expresses this idea as Something is needed by/for me.

Нужен is not a normal verb like need in English. It is the short form of the adjective нужный, meaning needed or necessary.

In this pattern:

• мне нужен = I need / it is needed by me
• тебе нужен = you need
• нам нужен = we need

So the core idea is adjectival: The translation is needed.

This is a very common Russian way to say that someone needs something.

Because нужен agrees with перевод, and перевод is:

• masculine
• singular

So the form must be masculine singular:

• нужен for masculine singular
• нужна for feminine singular
• нужно for neuter singular
• нужны for plural

Compare:

• Мне нужен перевод. = I need a translation.
• Мне нужна книга. = I need a book.
• Мне нужно слово. = I need a word.
• Мне нужны документы. = I need documents.

Точный describes перевод, so it must match it in gender, number, and case.

Перевод is masculine singular, and here it is the direct object of the whole expression, so you get:

• точный перевод

Also, for an inanimate masculine noun like перевод, the accusative form looks the same as the nominative form. So:

• nominative: точный перевод
• accusative: точный перевод

That is why nothing visibly changes.

Because Russian says translation of a word, and after перевод the thing being translated is usually put in the genitive.

So:

• перевод слова = translation of a word
• перевод этого слова = translation of this word

Here:

• это changes to этого
• слово changes to слова

Both words are in the genitive singular.

A useful question to remember is:

• перевод чего? = translation of what?

Answer:

• этого слова

In present-tense Russian, the verb to be is usually omitted.

So Russian does not normally say something equivalent to Мне есть нужен... That would be wrong.

Instead, Russian simply says:

• Мне нужен точный перевод этого слова.

This naturally means:

• I need an exact translation of this word.

The idea of is is understood automatically.

No, Russian word order is fairly flexible, though some orders sound more neutral than others.

Мне нужен точный перевод этого слова is a neutral, natural sentence.

You could also say:

• Точный перевод этого слова мне нужен.

This gives more emphasis to точный перевод этого слова.

So the meaning stays basically the same, but the focus can shift depending on word order.

It can cover all of those ideas depending on context.

In this sentence, точный перевод means something like:

• an exact translation
• a precise translation
• an accurate translation

A learner should understand that точный emphasizes correctness and precision. The speaker does not want just any translation, but one that matches the word very carefully.

Yes, but it is less natural in this everyday situation.

Я нуждаюсь в... means I am in need of... or I require..., and it sounds more formal, heavier, or more abstract.

Compare:

• Мне нужен точный перевод этого слова. = the normal, everyday way
• Я нуждаюсь в точном переводе этого слова. = more formal or stylistically marked

So for ordinary speech, мне нужен... is the better choice.

Singular is natural because the speaker wants one exact translation of this word.

Using the plural would change the meaning. For example:

• Мне нужны переводы этого слова. = I need translations of this word.

That suggests multiple translations, perhaps from different contexts or languages. But your sentence asks for one exact rendering, so singular перевод is the natural choice.

A very literal breakdown is:

• Мне = to me / for me
• нужен = is needed
• точный = exact / precise
• перевод = translation
• этого слова = of this word

So the sentence is literally:

To me is needed an exact translation of this word.

That literal version sounds unnatural in English, but it helps explain the Russian grammar.