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  "body": "\n## Definition\n\nLet $z, w \\in \\mathbb{C}$ with $z \\neq 0$.\nThe complex powers of $z$ with exponent $w$ are the numbers $e^{w \\ell}$, where $\\ell$ runs through the [[complex logarithms|0X2FY491PF]] of $z$.\nThe principal value of $z^w$ is\n$$z^w = e^{w \\operatorname{Log} z},$$\nwhere $\\operatorname{Log}$ is the [[principal branch|0YKY4M3KRQ]] of the logarithm and $e^u$ is the [[complex exponential|054VXHMCA3]] of $u$.\n\n## Remarks\n\nThe powers of $z$ with exponent $w$ are the numbers $e^{w \\operatorname{Log} z} e^{2 \\pi i k w}$ with $k \\in \\mathbb{Z}$. There is one value if $w$ is an integer, there are $q$ values if $w = p/q$ in lowest terms with $q \\ge 2$, and there are infinitely many values otherwise.\nFor an integer $w = n$, $z^n$ is the usual product of $n$ factors $z$, or of $-n$ factors $1/z$.\nFor example, $i^i = e^{i \\cdot i \\pi/2} = e^{-\\pi/2}$, a positive real number.\nThe usual rules can fail: with principal values, $(z^a)^b \\neq z^{ab}$ for some $z, a, b$. A common convention for $z = 0$ is $0^0 = 1$ and $0^w = 0$ for $w \\neq 0$.\n\n## Explanations\n\n### L1 human\n\nFor real numbers, $a^b = e^{b \\ln a}$ when $a \u003e 0$. The complex power uses the same formula with a complex logarithm.\nBecause the logarithm has many values, the power can have many values too, and the principal value picks one of them.\n",
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