Learn Arabic (Modern Standard): وضعت المرأة الملابس في النشافة. - The woman put the clothes in the clothes dryer.

It is built on the pattern فَعَّالَة (fa''āla), which is used for modern devices and machines. It takes the root ن - ش - ف (relating to drying or absorbing) and fits it into this shape. This is exactly the same pattern as سيارة (car) and ثلاجة (fridge).

Are there other everyday words from the same root as النشافة?

Yes, a very common related word is منشفة (towel). Notice how a manual towel uses an older, traditional Arabic pattern for hand-tools (مِفْعَلَة), while the electronic dryer uses the modern pattern for machinery and appliances (فَعَّالَة).

How do you pronounce the end of the verb وضعت when it connects to المرأة?

You pronounce it with a temporary "i" sound (a kasra): waḍa'ati-l-mar'a. The verb وضعت normally ends with a sukoon (a non-voweled consonant). Because the word المرأة also starts with a non-voweled consonant sound (the "l" of the "al-"), Arabic adds a kasra to the verb's ending to prevent two of these silent beats in a row.

Since الملابس is an inanimate plural, why doesn't it make the verb feminine singular?