Learn Russian: Мой брат работает пилой, и везде много опилок. - My brother works with a saw, and everywhere there is a lot of sawdust.

The dictionary form is опилки. This is a plural-only noun. Because it follows много (a lot of), it must be in the genitive case. To form the genitive plural, plural-only nouns ending in -ки usually drop the final vowel, giving us the zero-ending form опилок.

Why do we say работает пилой instead of работает с пилой?

Пилой is the instrumental case of пила (saw). When a noun is used as a physical tool to perform an action, Russian uses the bare instrumental case without a preposition. Adding the preposition с (с пилой) would mean your brother is working alongside the saw, as if it were a coworker!

Are the words пилой (saw) and опилок (sawdust) related?

Yes, they share the same root! Пила is the noun for a saw, and пилить is the verb for to saw. The prefix о- can indicate the remnants or byproduct of an action, so опилки literally translates to the shavings left over from sawing.

Since опилки is plural, why isn't there a plural verb meaning "there are" after везде?

In Russian, sentences stating that something exists or is present somewhere often completely omit the verb "to be" in the present tense. You simply say везде много опилок (literally "everywhere a lot of sawdust") instead of inserting a verb to represent "there is" or "there are".