The correct form is "jedem." This is the dative case form of the word "jeder," which means "each" or "every." It is used when referring to the indirect object of a sentence, such as "Ich gebe jedem ein Geschenk" (I give a gift to everyone). On the other hand, "jeden" is the accusative case form of "jeder," and is used when the word is the direct object of a sentence, such as "Ich sehe jeden Tag einen Film" (I watch a movie every day). **

"Jeden zweiten Tag" is a German phrase that translates to "every other day" in English. It indicates a frequency of occurrence where something happens on alternate days, skipping every second day in between. For example, if someone says they exercise "jeden zweiten Tag," it means they exercise every other day. **

"Jeden zweiten Sonntag" means "every second Sunday" in German. It indicates that an event or activity occurs on every second Sunday, or that a particular day of the week is repeated every two weeks. For example, if a club meets "jeden zweiten Sonntag," it means they meet every other Sunday. **

Without the specific context, it is difficult to determine whether "jedes" or "jeden" is the correct word to use. However, "jedes" is used for neuter nouns and "jeden" is used for masculine nouns in the accusative case. It is important to consider the gender and case of the noun in the sentence in order to determine the correct form to use. **

"Auf jeden Fall" is not written together because it is a prepositional phrase in German, where "auf" is a preposition and "jeden Fall" is the object of the preposition. In German, prepositional phrases are typically written as separate words, unlike in English where they are often written together. This is a common structure in German grammar and helps to clarify the relationship between the preposition and its object. **

The phrase "auf jeden Fall" is written correctly in German. It translates to "in any case" or "definitely" in English. It is used to express certainty or to emphasize a point. This phrase is commonly used in spoken and written German. **

The difference between "täglich" and "jeden Tag" lies in their formality and usage. "Täglich" is more formal and is often used in written or official contexts, while "jeden Tag" is more commonly used in everyday spoken language. Both phrases mean "every day," but "täglich" may be preferred in more professional settings, while "jeden Tag" is used in casual conversations. **

"Auf jeden Fall" is not written together because it is a prepositional phrase in German, consisting of the preposition "auf" (on) and the noun phrase "jeden Fall" (every case). In German, prepositional phrases are typically written separately, unlike in English where they are often written together as one word. Therefore, "auf jeden Fall" remains separate in German to maintain proper grammar and syntax. **