Graph of the Power Function $y = x^{2/3}$

$y = x^{2/3}$ is a power function with a fractional exponent, and can be written $x^{2/3} = \left(\sqrt[3]{x}\right)^2 = \sqrt[3]{x^2}$. Combining a cube root with a square, and because $x^2$ is always non-negative, it is defined as a real number even for negative $x$.

Its domain is all real numbers and its range is $y \geq 0$; the squaring keeps the value from ever being negative. Moreover $(-x)^{2/3} = \sqrt[3]{(-x)^2} = \sqrt[3]{x^2} = x^{2/3}$, so it is an even function, symmetric about the $y$-axis.

The derivative is $y' = \dfrac{2}{3}x^{-1/3} = \dfrac{2}{3\sqrt[3]{x}}$. As $x \to 0^{+}$, $y' \to +\infty$, and as $x \to 0^{-}$, $y' \to -\infty$, so the tangents on both sides become vertical at the origin. Such a sharp point is called a cusp. In contrast to the parabola $y = x^2$, which is smooth at the origin, $x^{2/3}$ plunges into the origin even more sharply than a V.

It decreases for $x < 0$ and increases for $x > 0$, taking its minimum value $0$ at the origin $(0, 0)$, and is smooth everywhere else. Concretely, $y = 1$ at $x = \pm 1$, $y = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ at $x = \pm 8$, and $y = 3^2 = 9$ at $x = \pm 27$.

As a power function $x^p$ with $0 < p < 1$, it rises steeply near the origin and gently far away. This shape appears in the famous astroid:

Two-thirds-power relationships also occur in natural scaling laws, such as Kepler's third law (the square of the orbital period is proportional to the cube of the semi-major axis).