How exactly is the function $x^a$ defined?

Common convention is

  • no problem with positive basis (and the log representation is fine);

  • no problem with zero basis and positive exponent;

  • $0^0$ can be $0$ or $1$ depending on contexts; negative power not allowed;

  • if the basis is negative,

    • a rational exponent must be written in simplified form and have an odd denominator. The rule $x^{m/n}=\sqrt[n]{x^m}=(\sqrt[n]x)^m$ works.

    • an irrational exponent is not allowed.

If you allow complex answers, then

  • rational powers define $n$ distinct branches,

  • irrational powers define a principal branch $\sqrt[m/n]{-x}\text{ cis}(\frac{m\pi}n)$ or are not allowed.

Tags:

Exponentiation

Calculus

Real Analysis

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