Zam will cut soft materials like malachite and turquoise.
If you use a hard felt buff with zam you should be able to
remove the sratches. It will also leave a good polish.
Malachite is really easy to repolish. White diamond tripoli
will smooth it, if your other compounds haven’t done so already.
Zam may be all you need for that. It will polish pretty well too.
For a super polish, use a little Red rouge on a soft muslin
wheel, just don’t use too much, and use a light touch. Be sure
not to work it longer than needed, or it will undercut along the
banding, giving you a banded surface texture. Will still have
the high polish, though, and sometimes that slight texture is