Arithmetic roots of powers with a natural exponent

The n-th arithmetic root (n ≥ 2 a natural number) of a nonnegative number a is the nonnegative number whose n-th power equals a; it is written ⁿ√a. So ⁿ√a = b means b ≥ 0 and bⁿ = a; for example ⁴√81 = 3 because 3 > 0 and 3⁴ = 81. From the definition, for a ≥ 0 we get (ⁿ√a)ⁿ = a and ⁿ√(aⁿ) = a. For n = 2 the root is the square root, for n = 3 the cube root. In xⁿ = a with n even and a > 0 there are two roots, x = ±ⁿ√a; if n is odd there is exactly one root whatever the sign of a, for example x³ = -8 gives x = -2. An even root of a negative number does not exist. The properties ⁿ√(ab) = ⁿ√a · ⁿ√b and ⁿ√(a/b) = ⁿ√a / ⁿ√b (a ≥ 0, b > 0) also hold.