Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive real number.
A D993: Real number $a \in \mathbb{R}$ is a power mean of $x_1, \dots, x_N$ with respect to $p \in \mathbb{R} \setminus \{ 0 \}$ if and only if \begin{equation} \sum_{n = 1}^N a^p = \sum_{n = 1}^N x^p_n \end{equation}

Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive real number.
A D993: Real number $a \in \mathbb{R}$ is a power mean of $x_1, \dots, x_N$ with respect to $p \in \mathbb{R} \setminus \{ 0 \}$ if and only if \begin{equation} \underbrace{a^p + a^p + \cdots + a^p}_{N \text{ times}} = x^p_1 + x^p_2 + \cdots x^p_N \end{equation}