Restart router while range extender connected

> Simple question, but one I can't find the answer to: if I restart my
> router will my range extender automatically reconnect?

   It should.  One reason you can't find the answer is that it's a "Does
gravity always pull down?" kind of question.  This stuff would not be
usable if any minor disturbance forced the owner to reconfigure
everything.

> [...] I'm experiencing slow internet and want to restart but I'm not
> sure if I'll have to reconfigure the extender.

   The settings on your (unspecified) extender should not be affected by
anything which happens to the router.  Nor by a loss of power to the
extender.  You should need to deal with the extender's web interface or
its reset button to affect its settings.

   I can't guarantee that nothing will ever go wrong, but restarting
your router should be pretty safe.  (Only one way to find out.)