Learn French: Elle a rangé les sèche-cheveux en nettoyant la salle de bain. - She put away the hairdryers while cleaning the bathroom.

This is a compound noun made of a verb (sèche) and a noun (cheveux). In French, the verb part of a compound noun never takes a plural marker. The noun part usually takes an -s or -x in the plural, but since cheveux already ends in -x, the whole word looks exactly the same in singular (un sèche-cheveux) and plural (les sèche-cheveux).

Why is en nettoyant used here instead of a phrase like pendant que?

en nettoyant is the gérondif (gerund). It is formed with en plus the present participle of the verb. It is the most natural way to show that the subject is doing two actions at the exact same time. You could say pendant qu'elle nettoyait, but the gérondif is much more concise and common in this context.

I sometimes see salle de bain written with an s at the end of bain. Which one is correct?

Both are completely correct! Traditionally, it was written salle de bains (a room for baths). However, since the 1990 spelling reform, salle de bain (a room with a bath) is also officially accepted. You will see both spellings used frequently by native speakers.

What is the exact difference between ranger and nettoyer in this sentence?