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The present tense of "bathe" is "bathing" - still pronounced with a long A as in bathe.
The verb of bath is bathe.
As in "to bathe somewhere" or "to bathe in something".
It wouldn't be the best idea. Your kitten may get an allergy from it. Visit your local pet store and inquire what shampoo would suit best
"I will bathe".
Bathe is correct.