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Apagarse vs. extinguirse

The Spanish verbs apagarse and extinguirse both translate to to go out or to be extinguished in English, but they differ in tone and context. Apagarse is the versatile, everyday word used for lights, small fires, and electronic devices. Conversely, extinguirse is a more formal or dramatic term used for large blazes being entirely consumed, species dying out completely, or something coming to a definitive end.

Apagarse

A2
The word apagarse is used in everyday language to describe when a light, electronic device, small fire, or machine goes out, turns off, or loses its intensity.
La luz de la cocina acaba de apagarse de repente.
(The light in the kitchen just went out suddenly.)
Mi computadora portátil va a apagarse porque no tiene batería.
(My laptop is going to turn off because it has no battery.)
El fuego de la chimenea comenzó a apagarse lentamente durante la noche.
(The fire in the fireplace slowly began to go out during the night.)
La pantalla de la televisión suele apagarse después de diez minutos de inactividad.
(The television screen usually turns off after ten minutes of inactivity.)
Su entusiasmo pareció apagarse cuando escuchó los resultados finales.
(His enthusiasm seemed to fade away when he heard the final results.)
La vela va a apagarse si dejas la ventana abierta con este viento.
(The candle is going to go out if you leave the window open with this wind.)
Las farolas de la calle siempre vuelven a apagarse cuando amanece.
(The street lamps always go out again when dawn breaks.)
El motor hizo un ruido extraño antes de apagarse por completo en medio de la carretera.
(The engine made a strange noise before turning off completely in the middle of the road.)

Extinguirse

B2
The word extinguirse is a formal term used to describe a fire that is definitively and completely put out, often on a large scale, or something that is utterly consumed, destroyed, or dies out permanently.
El enorme incendio forestal tardó tres días enteros en extinguirse.
(The massive forest fire took three whole days to completely go out.)
La pequeña llama del fósforo terminó por extinguirse antes de que pudiera encender la estufa.
(The small match flame ended up going out before he could light the stove.)
Los valientes bomberos trabajaron sin descanso hasta que las llamas lograron extinguirse.
(The brave firefighters worked restlessly until the flames managed to be extinguished.)
Los biólogos temen que esta especie de tortuga marina pueda extinguirse pronto.
(Biologists fear that this species of sea turtle may die out soon.)
El contrato de arrendamiento está a punto de extinguirse a finales de este mes.
(The lease agreement is about to expire at the end of this month.)
La esperanza de encontrar a los montañistas perdidos comenzó a extinguirse al cuarto día.
(The hope of finding the lost mountaineers began to die out on the fourth day.)
El volcán dejó de hacer erupción y su actividad interna parece extinguirse gradualmente.
(The volcano stopped erupting and its internal activity seems to be dying out gradually.)
Cuando se consuma todo el oxígeno del recipiente, la reacción química va a extinguirse.
(When all the oxygen in the container is consumed, the chemical reaction will be extinguished.)

Summary

To summarize, choose apagarse for your standard, daily situations when a light goes out, a screen turns off, or a small flame dies down. Reserve extinguirse for grander, formal, or scientific contexts, such as an uncontrollable blaze finally being put out, or when a species, legal right, or intense emotion completely and permanently ceases to exist.