When we write "the sixth root of a5" as , the 6 is the INDEX.
We are assuming that all the letters stand for positive numbers so that
is the principle square root, etc.
Radicals, like exponents, do distribute across multiplication and division, BUT they do NOT distribute across addition and subtraction.
Simplifying radicals used the multiplication property above to factor out any perfect squares, cubes, etc.
From the last section: so the same power and root "undo" eachother.
Again, considering all the variables to be positive:
To simplify square roots, remove all perfect squares. For square roots of numbers, this means to look for:
12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, and so on.
For cube roots, look for perfect cubes. Such as:
13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125, and so on.