"Please take a shower immediately."
The reason is because the plain form of the verb 「浴びます」is, in fact, 「浴びる」. Since it’s a ru-verb, it takes off the final 〜る and adds 〜て in te-form. I feel like Duolingo should have introduced plain verbs earlier to make a clarification and avoid confusion.
The plain form itself is the dictionary form and is used in casual conversations, cartoons, books etc. and is more than needed. 僕はあなたを手伝ったことを願う。他の質問があったら、聞いてください。(^_^) (i.e. I hope that I helped. If there are other questions, please ask.)
Your phrase was wrong not because of kanji, but wrong conjugation of verb.
浴びる is an 一段動詞 (ichi-dan verb; a verb with only 1 stem: あび), its te-form is 浴びて (not ×
帰る【かえる】on the other hand is a 五段動詞 (go-dan verb, having 5 stems (for the 5 vowels)) : かえら, かえれ, かえり, かえろ, かえる); the u-stem is used in the plain form (dictionary form): 帰る, the a-stem for the plain form negative: 帰らない, the i-stem used withe politeness sufix: 帰ります; and, indeed, the te-form is 帰って (and plain past 帰った), not ×
かえりて nor × かえて
No, because 浴びる 「あびる」is an -iru/-eru verb - it ends with an iru or an eru sound, it rhymes with -iru or -eru.
iru/eru, drop the ru.
co conjugation is done by dropping the -ru from the end of the verb and adding whichever suffix you need.
You get the point. -nde ending is added to regular verbs (godan verbs), which end with -mu, -bu, -nu, which abiru is not. If we were talking about some theoretical "abu" verb, then you would say "abinde". But here the dictionary/plain form is "abiru", so you conjugate it according to the -iru/-eru rule. IRU ERU DROP THE RU.