Akademik

как же
КАК ЖЕ coll
[Particle]
=====
1. Also: КАК ЖЕ, КАК ЖЕ coll [used as an affirmative response to a question or in support of a statement]
certainly, it is clear or agreed (without needing to be stated or proven):
- (yes (but, why),) of course;
- of course I do (will etc);
- yes, indeed (I do etc);
- sure(ly);
- naturally;
- [in limited contexts] well then (, well then);
- [in response to "Do you mind...?", "Do you object to...?" etc] of course not (, of course not).
     ♦ "Может быть, помните? Я вам рассказывал об Абуталипе Куттыбаеве". - "А, ну как же, как же! Прекрасно помню" (Айтматов 2). "Perhaps you'll remember - I told you once about Abutalip Kuttybaev." "Oh, yes, of course, indeed I do! I remember well now" (2a).
     ♦ "Ах да, конечно же, "Капитаны", "Африка", "Нигер"... Как же, как же, в наше время все гимназисты от этих стихов [Гумилёва] с ума сходили" (Копелев 1). "Ah yes, of course, 'Captains/ Africa,' The Niger.' But of course, in our day all the Gymnasium students were crazy about some of his [Gumilev's] poems" (1a).
     ♦ "Разве вы ездите верхом?" - "Как же! К нынешнему дню и фрак нарочно заказывал" (Гончаров 1). "But do you ride?" "Of course I do! I had the coat specially made for to-day" (1a).
     ♦ "У меня для вас письмо от сына, от Алексея Сергеевича". - "А-а-а! Как же, как же!" (Федин 1). "I have a letter for you from your son, from Aleksei Sergeyevich." "A-A-ah! Well then, well then" (1a).
     ♦ "Мне нужно кое о чем переговорить, так не хотите ли заехать ко мне?" - "Как же, как же", - сказал поспешно Хлобуев и вышел с ним (Гоголь 3). "There's something I'd like to discuss with you. Would you mind coming to my house?" "Of course not, of course not," Khlobuyev said hastily and went out with him (3a).
2. used to express disagreement, doubt that sth. said by the interlocutor is as stated (the speaker often repeats the part of the interlocutor's statement with which he disagrees):
- that's what you think!;
- (oh) sure!;
- yeah, right!;
- [in limited contexts] sure one does <will etc>.
     ♦ [Катерина:] И на воле-то он словно связанный. [Варвара:] Да, как же, связанный! Он как выедет, так запьет (Островский 6). [К.:] So, even on his own, it's just as if he were still tied to her. [V.:] That's what you think! As soon as he leaves here, he'll start in drinking (6f).
     ♦ "Ленту, что ли, привезли? - спросил я. - Опять "Девушку с гитарой'?" - "Как же, ленту, дожидайся!" - ответил Чудаков (Аксёнов 1). "What, did you bring a movie?" I asked. "The Girl with the Guitar again?" "Oh sure, a movie, that's a good one!" answered Chudakov (1a).
     ♦ "Ты думаешь, ты любишь?! Как же! Да ты за человека никого не считаешь" (Битов 2). "You think you love? Sure! But you don't consider anyone a person" (2a).
     ♦ [Бусыгин:] Он говорил, что он сам сочиняет музыку. [Нина (насмешливо):] Ну как же (Вампилов 4). [В.:] He said he composed music himself. [N (ironically):] Sure he does (4a).

Большой русско-английский фразеологический словарь. — М.: ACT-ПРЕСС КНИГА. . 2004.