Mountains have roots, but there is only one instance of “mountain-roots”.

• 12.021 he shook the mountain-roots.

Here are other forms of this concept:

• 05.012 down at the very roots of the mountain.
• 12.012 right at the Mountain’s root.
• 12.096 beneath the roots of the Mountain,

The Roots of the Mountains: Wherein is Told Somewhat of the Lives of the Med of Burgdale, their Friends, Their Neighbors, Their Foemen, and Their Fellows-in-Arms, by William Morris, is the earliest that I know where to look for this idea.  Famous Tolkien scholar Douglas A. Anderson reported that “Lewis writes that Tolkien ‘grew up on William Morris and George MacDonald.'”  I am about to send a note to that eminent gentleman to ascertain his source.  Update just two hours later – Dr. Anderson has kindly replied that “the quote comes from a CS Lewis letter to Arthur Greeves, dated 4 February 1933. It’s in They Stand Together (p. 449), and also (I suppose) the Collected Letters of CSL.”  I will endeavor to put my hands on one book or the other forthwith.

Meanwhile, “mountain-root” by any spelling is not found in OED, so we will give it the JRRT tag.  And by “we”, I mean “the dogs and I”.

Anderson, Douglas A. Personal correspondence. September 19, 2017. email.

Anderson, Douglas A. “Week 3 Friday Lecture.” Roots of the Mountain.  Course for Signum University, lecture delivered September 12, 2014. Web.