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?