Learn German: Ich nehme beide teuren Exemplare. - I am taking both expensive copies.

In the plural, the word beide (both) acts just like the definite article die (the). Because of this, any adjective following it gets the weak ending -en. If you used a word like viele (many) instead, it would act like there is no article, and the adjective would end in -e (viele teure Exemplare).

Does the word beide change its ending depending on the case?

Yes, beide declines exactly like the plural definite article. In this sentence, it is the direct object (accusative case), so it takes the -e ending just like die would.

Why is it spelled teuren and not teueren?

When the adjective teuer (expensive) gets any ending, it drops the unaccented e before the r to make it easier to pronounce. So teuer plus the -en ending becomes teuren.

Could I use the word Kopien instead of Exemplare for copies?

Usually not in this context. Exemplare refers to individual physical items of a publication, like copies of a book, newspaper, or magazine. Kopien specifically means photocopies or exact duplicate reproductions.