Её улыбка выражает радость.

Because её means her in the possessive sense: her smile.

• она = she
• её = her

So:

• она улыбается = she is smiling
• её улыбка = her smile

In this sentence, её is showing possession, so её is the correct word.

Улыбка is in the nominative case because it is the subject of the sentence — the thing doing the action.

In Её улыбка выражает радость, the smile is what expresses the joy.

So:

• улыбка = subject
• subject in Russian usually takes the nominative

That is why it stays as улыбка.

Выражает is the 3rd person singular present tense form of выражать (to express).

It matches the subject:

• улыбка = smile
• singular
• so the verb must also be singular: выражает

Compare:

• улыбка выражает = the smile expresses
• улыбки выражают = the smiles express

So the ending -ет / -ает here shows he/she/it expresses.

Yes, радость is the direct object, so it is in the accusative case.

But for this noun, the accusative singular looks the same as the nominative singular.

So:

• nominative: радость
• accusative: радость

That is why you do not see a visible ending change here.

This is normal for many Russian nouns: sometimes the case changes the form, and sometimes it does not.

Russian uses the imperfective verb выражать here because the sentence states a general fact or a continuous characteristic:

• Her smile expresses joy.

The idea is not about one completed act, but about what her smile shows in general.

If you used the perfective verb выразить, its present-tense forms actually refer to the future:

• выразит = will express

So:

• выражает = expresses / is expressing
• выразит = will express

That is why выражает is the natural choice here.

In this sentence, её is a possessive pronoun meaning her.

A useful point: его, её, and их do not change form the way many adjectives and pronouns do.

So even though улыбка is feminine singular, you still say:

• её улыбка = her smile
• её лицо = her face
• её глаза = her eyes

The form её stays the same.

Yes. Russian word order is more flexible than English word order.

The neutral, standard order here is:

• Её улыбка выражает радость.

But other orders are possible for emphasis, style, or focus, for example:

• Радость выражает её улыбка.
• Выражает радость её улыбка.

These can sound more literary or marked, depending on context.

For a learner, the safest default is the normal subject–verb–object order:

• Её улыбка
  • выражает
    • радость

Её is pronounced roughly yi-YO.

A few useful points:

• the stress is on ё
• ё always represents the yo sound
• in many Russian texts, people write её as ее without the dots

So you may see both:

• её
• ее

But when spoken, it is still pronounced её.

The dots are important for learners because they show the pronunciation clearly.

Not exactly.

Радость usually means joy or gladness — a feeling of delight or pleasure.

It is often more immediate or emotional than happiness.

For comparison:

• радость = joy
• счастье = happiness

So in this sentence, радость is a very natural choice if the idea is that the smile shows a feeling of joy.

Russian does not have articles.

So where English says:

• her smile
• the smile
• a smile

Russian simply uses the noun, and the meaning is understood from context:

• её улыбка

Whether English translation uses a, the, or no article at all depends on the situation, not on a separate Russian word.

Yes, depending on context, the Russian present tense can sometimes correspond to either:

• Her smile expresses joy
• Her smile is expressing joy

However, in a simple standalone sentence like this, English usually prefers:

• Her smile expresses joy

That sounds more natural as a general description.

So the Russian present tense is somewhat broader than English, and context decides the best translation.