Potential forms are used to describe that you are capable of doing something or that something is possible.

The short potential form

The ren’youkei + える (eru) is the short potential form and can only be used with Group 4 (yodan katsuyou) verbs.

This combination has a contraction where the final “i” from the ren’youkei form is dropped.

The contraction works as follows:
買い (kai = ren’youkei) → 買い+える(kai +eru) → 買える (kaeru = short potential form)

Ren'youkeiRen'youkei + える (eru)
買い kai買える kaeru
書き kaki書ける kakeru
泳ぎ oyogi泳げる oyogeru
出し dashi出せる daseru
待ち machi待てる materu
死に shini死ねる shineru
遊び asobi遊べる asoberu
飲み nomi飲める nomeru
入り hairi入れる haireru

The addition える (eru) comes from the verb 得る (eru), meaning “to acquire”. The combination of 書く (kaku), meaning “to write”, and 得る (eru), meaning “to acquire”, would translate to “to acquire the writing” or “to be able to write”.

When adding 得る (eru), a Group 2 verb, to the ren’youkei the resulting verbs are also conjugated as Group 2 verbs.

Nihon de Doitsu no kuruma wo kaemasu ka.
Can you buy German cars in Japan?

Pen ga attara kaketa.
If I had a pen, I would have been able to write it.

There are a few verbs without contractions:
あり得る (arieru) and 起こり得る (okorieru)

Sore ga arienai!
That can’t be! / That’s impossible!

Kimi ni mo okorieru.
It can happen to you too.

The long potential form

The long potential form is created by placing ことができる (koto ga dekiru) after the rentaikei conjugation. This construction can be used with both Group 2 (nidan katsuyou) and Group 4 (yodan katsuyou) verbs.

Hashi de taberu koto ga dekimasu.
I can eat with chopsticks.

Nihongo mo Arabiago mo kaku koto dekimasu.
I can write both Japanese and Arabic.

Note: In the spoken language particles are sometimes left out, as shown in the second example.

The passive form

The potential passive form is constructed in the same way as the normal passive form, but the grammatical subject of the sentence is usually separated by the particle は (wa). This form is often used to create a potential form from Group 2 (nidan katsuyou) verbs.

Yuufoo ga mirareru!
I can see a UFO!