Learn Finnish: Uudet opiskelijat muuttavat farkut kauniiksi laukuiksi. - The new students turn the jeans into beautiful bags.

Laukuiksi is in the translative plural case, meaning "into bags". Adding the translative ending -ksi closes the syllable, which triggers consonant gradation. The strong kk in laukku weakens to a single k. You add the plural -i- and the translative -ksi to this weak stem to get laukuiksi.

Is the adjective kauniiksi singular or plural here?

It is plural to agree with laukuiksi, but interestingly, the singular and plural translative forms of kaunis look exactly the same! The singular stem is kaunii- plus -ksi (kauniiksi). For the plural, the plural marker -i- simply merges with the long ii of the stem, also resulting in kauniiksi.

Why is the object farkut in the nominative plural instead of the partitive?

Farkut (jeans) is a plural-only word in Finnish. Because the students are transforming the entire pair of jeans (a completed action), it takes the total object form. For plural words, the total object is identical to the nominative plural. If they were only modifying part of the jeans, it would be in the partitive (farkkuja).

How does the verb muuttaa dictate the case of the words that follow it here?