I think "must" and "need" only have different meanings when used with a negative. (ex. you must not be out past 7/you do not have to be out past 7: must not = prohibited; do not have to = not obligated)
The technical difference I found between the two is apparently that "must" implies opinion whereas "have to" implies an obligation, but I think English has strayed from that. Where I am, the only difference is that "must" seems more serious, like there is a real obligation with consequences, while "have to" has been weakened by being used figuratively (ex. "You have to try this!").
If other figures of speech, colloquial terms, and synonyms are accepted in other parts of the course than these should be interchangeable in positive statements.
There is no Perfective present tense. There's a reason for that: It's more complicated than the following, but...
"The aspects are:
"Imperfective - Incomplete, ongoing, habitual, reversed or repeated actions
"Perfective - Actions completed successfully.
In the present tense, things are on-going, so they can't have been completed. Perfective present tense would be self-contradictory: an ongoing thing which has been completed makes no sense.
Things obviously can be completed in the past, and also can be completed in the future, so Perfective tenses exist for those time frames. Imperfective covers all the time frames.
Not a native here but this is not how I understand it. The verb помыть here is in the infinitive form. Meaning, it is the base from which other forms are derived (in Spanish this would be -er or -ire in Italian, while in Turkish it's -mAk...etc).
On a side note, the terms perfective and imperfective is quite misleading in my opinion. From what I've figured out following some videos on Youtube, it turns out that these differences are simply related to the type of the present (and surely it reflects back on the past tense as well). I find the most suitable (almost) equivalent to this system is in Turkish where in this language there are 2 present tenses; In one of them the verb ends with -Ar, while for the other it ends with -Ayor. In the Turkish program on Duolingo, they equalize this to regular present tense (e.g. I eat) while the second one is equated to the continuous (e.g. I am eating). Projecting this understanding, I think, helps me a lot in understanding the perfective and imperfective in Russian. However I read about it (specially on Duolingo) I never quite understood that until I watched some videos and recalled my knowledge of Turkish and English. It's just about using the verb to express a general or habit tense (imperfect), while the other one is about a continuous or current action at the moment (perfective). Hope this helps.
I found this sentence very difficult, with several bits that I didn't understand. I'm going to sum up what I figured out thanks to various comments from other users above and from the great internet. In no particular order:
*1. Three plates: *
1.1 declension of noun with number:
"The nominative case of два (две), три, четыре and of the compound numbers ending in 2, 3, 4 (23,134, etc.) is followed by the genitive singular of nouns" (http://russianlearn.com/grammar/category/cardinal_numerals).
1.2 Genitive of plate:
таре́лка is feminine inanimate (https://en.m.wiktionary.org/wiki/%D1%82%D0%B0%D1%80%D0%B5%D0%BB%D0%BA%D0%B0#Russian). Normally, that would make the genitive singular ending ы, but "After a sibilant or a velar (г, к, or х) consonant, и is written." (https://en.m.wikipedia.org/wiki/Russian_declension).
2. On the verb form (source for the following is German PONS cheatsheet for Russian):
Imperfective aspect describes duration/ repetition/ habit of an action. The point is the doing. Perfective aspect describes a specific, singular action/ its completion/ a chain of such events. The point is that the action has been completed/ the result achieved. To transform an imperfective verb into a perfective one, you add a prefix; in our case "по". Other prefixes exist for other verbs. Is there a rule to know which prefix applies? (Side note: Perfective verbs also exist. To go the other way, you add a sufix. So how do you know which vetb is naturally perfective/ imperfective?)
3. Мне надо:
надо (like нужнo) belongs to the category of predicative adverbs. "Syntactically, these adverbs express the predicate of impersonal sentences. The logical subject (when there is one) stands in the dative case" (http://russianlearn.com/grammar/category/predicative_adverbs). So мне must be dative here (rather than prepositional).
3.2 надо as opposed to нужно
3.2.1 Don't use надо when what you need is a noun. "In cases when your sentence states that you need to do something, надо and нужно are interchangeable."; however, the latter is more formaland moreover declined (in accordance with the noun). (https://russian.stackexchange.com/questions/1105/usage-patterns-of-надо-vs-нужно#1232).
3.2.2 some idiomatic expressions
where usage suggests preferring one over the other, though both are correct, e.g. надо бы.
Try this link . I found it useful. http://www.russianlessons.net/lessons/lesson2_main.php
The subject of the sentence is more "three plates" - "Three plates need to be washed by you". That's why "you" is in dative.
This would be more obvious if it were just "You need to wash the plate" = Тебе надо помыть тарелка. In this sentence, "plate" is clearly in nominative case, making it the subject of the sentence.
I see you're studying Spanish - the sentence operates something like gustar - where things are like by you, but the sentence is translated as "you like the things".