What is the ruling regarding prayers performed with wide sleeves that exposed part of the arm, with a part exposed below the chin, and with thin socks that revealed the skin color? Are these prayers considered valid, and does the prayer become nullified by the appearance of a small part of the awrah, whether known or unknown?

The uncovering of the 'awrah (private parts) during prayer, if it is minor, does not invalidate it. However, if it is extensive, it invalidates the prayer according to the majority of scholars. The definition of "minor" is determined by custom. The forearms must be covered even if the sleeve is wide. Some scholars, like the Malikis, hold that the uncovering of the light 'awrah, such as the forearms, does not invalidate the prayer, though it is disliked, and the prayer should be repeated within its prescribed time due to its uncovering. Based on this latter opinion, the past prayers are valid. If the uncovering was extensive, scholars differ on the obligation of repeating the prayer for someone who was ignorant. Sheikh al-Islam Ibn Taymiyyah and some scholars are of the view that there is no obligation to repeat the prayer for someone who unknowingly omitted a condition. This opinion is strong, and there is no harm in acting upon it. However, if one wishes to be cautious, making up the prayers is permissible.