甲斐(かい) is a noun that signifies a 'reward', 'benefit', or 'payoff'. It is attached to verbs or nouns to express that an action (A) was worthwhile or rewarding. This grammar point is frequently written in hiragana as either かい or がい, with the form depending on the word it follows. It can be translated into English as 'it is rewarding to do (A)', or '(A) was well worth the effort'.

かい is typically followed by the particle が and the verb ある, literally meaning 'there is a reward'. The particle が is sometimes omitted in casual speech. When used with verbs, がい will connect to the ます-stem, while かい can follow any standard verb form (dictionary, past, etc.).

• 毎日(まいにち)一生懸命(いっしょうけんめい)勉強(べんきょう)して、試験(しけん)に合格(ごうかく)できた。がんばった甲斐(かい)があった。
  I studied hard every day and passed the exam. All that effort was worth it.
• 子(こ)どもたちの笑顔(えがお)を見(み)ると、働(はたら)いている甲斐(かい)があるなあと感(かん)じる。
  When I see my children's smiling faces, I feel that working is worth all the effort.
• 彼(かれ)は全(ぜん)然(ぜん)アドバイスを聞(き)かないから、手伝(てつだ)いがいがない。
  He never listens to advice, so helping him is not rewarding.

The main nuance between these two forms is that かい often puts more emphasis on the timing of the action based on the verb's tense, while がい tends to carry a feeling that something is 'inherently rewarding' or 'always worthwhile'.

When connecting to a noun, the particle の is necessary before かいがある.

• 何(なん)ヶ月(げつ)も続(つづ)けた練習(れんしゅう)の甲斐(かい)があり、ピアノコンクールで優勝(ゆうしょう)できた。
  The months of practice paid off, and I was able to win the piano competition.
• 厳(きび)しいトレーニングの甲斐(かい)があり、マラソンを完走(かんそう)できた。
  The tough training was well worth it, and I was able to finish the marathon.

Conversely, かいがない can be used to express that there is 'no point' in doing something or that it is 'not worth the effort'.

• うわー、つまらない映画(えいが)だ。ここまで歩(ある)いてきた甲斐(かい)がない。
  Ugh, what a boring movie. It wasn't worth walking all the way here for this.

かいがある is most often used when the speaker wishes to express that a positive outcome has been achieved as a result of enduring some kind of difficulty or hardship.