-(으)므로 means "because" or "since" - formal/written causation.
Formal reason - 위험하므로 주의하세요 (be careful because its dangerous)
Written style - Documents, announcements, formal speech
Cause-effect - More formal than -아/어서
See examples below for varied real-world usage.
Written/formal only - Rarely in casual speech
Similar to -(으)니까 but more formal
Since theres no time, please hurry.
Because tomorrow is a public holiday, we are closed.
Since the roads are slippery, please be careful when driving.
Because photography is prohibited inside the museum, we ask for your cooperation.
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