We have already met one way to make a suggestion.

It’s better that you quit smoking.

べき expresses a similar meaning, but is much more forceful—to the point where it is almost an order. While you may hear it in speech it is a considerably hard expression and is more often used in writing.

You should quit smoking.
You should keep promises.
If your leg hurts I think you should have the doctor take a look at it.

When used with the verb する it is usually abbreviated.

するべき → すべき

Seriously discuss with parents whether or not you should study abroad.

Unlike はず no particle is required to modify a noun.

A promise you should definitely keep.

べき can also be used to express a likely result, as well.

Surprising discovery.

べき can be changed to the negative by conjugating the auxiliary verb.

Even if there aren’t any cars you shouldn’t cross the road when the lights are red.
This town isn’t safe and so at night you shouldn’t leave the house.

A more literary way of forming the negative is with べきからず.

In England you should not drive at over 70mph on the motorways.

Incidentally, べし is the archaic form of べき. You may occasionally see it writing, but it is rarely ever used in speech.

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