nth Roots & Rational Exponents

A fractional exponent is a root.

x^(1/2) = √x, x^(1/3) = ∛x, and generally x^(1/n) is the nth root.

With a numerator too: x^(2/3) = ∛(x²), or equivalently (∛x)². Denominator gives the root, numerator gives the power. Taking the root first usually keeps the numbers smaller.

8^(2/3): cube root of 8 is 2, then squared is 4.

The advantage is that all the exponent rules you already know now apply to radicals. √x · ∛x becomes x^(1/2) · x^(1/3) = x^(5/6), which is far easier than manipulating the radicals directly.

Even roots of negatives are still undefined in the reals; odd roots are fine.

Define a^{1/n} = ⁿ√a as the principal nth root, then a^{m/n} = (a^{1/n})^m = (a^m)^{1/n}. This definition is forced by requiring the power rule (a^{m})^{n} = a^{mn} to hold for rational exponents, so it is the unique consistent extension.

Domain conditions depend on parity of the index. For odd n, ⁿ√a is defined for all real a. For even n it requires a ≥ 0, and ⁿ√(aⁿ) = |a|. This is why √(x²) = |x| but ∛(x³) = x.

Once in exponent form, every exponent law applies unchanged, which is the practical reason for the notation: expressions mixing different indices are simplified by converting to a common fractional exponent.

The extension continues to irrational exponents by continuity, giving meaning to 2^{√2} and e^x, which is what makes exponential functions well defined over all of ℝ.