The square root is the second root. The fourth root is the fourth root. There is no "fourth square root". (Your book and instructor aren't teaching you the notation and terminology solely to be annoying. If you don't know the lingo, you can't communicate with others, and this makes it very difficult for those others to help you. You are asking volunteers to give of their time; please have the courtesy to speak clearly. Thank you.)
I will guess (since grouping symbols are still not being used) that everything on either side of the equation is supposed to be included inside the radicals, so we have:
. . . . .4th-rt(x + 6) = 4th-rt(2x - 8)
Raise both sides to the fourth power to cancel the radicals (since a fourth power is the inverse of a fourth root), and solve the resulting linear equation:
. . . . .x + 6 = 2x - 8
(The process and answer will, of course, be quite different if my initial assumption was incorrect.)
Eliz.