Graph of the Fourth Root Function $y = \sqrt[4]{x}$

$y = \sqrt[4]{x}$ is the non-negative number whose fourth power is $x$, also written $x^{1/4}$. Containing a radical, it is called an irrational function, and can be seen as taking the square root twice: $\sqrt[4]{x} = \sqrt{\sqrt{x}}$.

Because $4$ is even, negative numbers have no real fourth root, so the domain is $x \geq 0$ and the range is $y \geq 0$. Larger inputs give larger outputs, so it is monotonically increasing, though it rises ever more gently.

The derivative $y' = \dfrac{1}{4}x^{-3/4}$ tends to $+\infty$ as $x \to 0^{+}$, so the tangent at the origin is vertical and the curve then leans over to the right. The second derivative is negative, so the graph is concave (bulging upward).

Compared with $\sqrt{x}$, for $x > 1$ we have $\sqrt[4]{x} < \sqrt{x}$ (an even flatter curve), while for $0 < x < 1$ we have $\sqrt[4]{x} > \sqrt{x}$ (closer to $1$). The two agree at $x = 0$ and $x = 1$.

Here are some values; each time $x$ is multiplied by $16$, $y$ doubles.

• $x = 1 \Rightarrow y = 1$
• $x = 16 \Rightarrow y = 2$
• $x = 81 \Rightarrow y = 3$
• $x = \dfrac{1}{16} \Rightarrow y = \dfrac{1}{2}$

It is the inverse of $y = x^4$ (for $x \geq 0$), the reflection of $y = x^4$ across the line $y = x$, and a member of the family $x^{1/n}$. In applications, the Stefan–Boltzmann law makes temperature proportional to the fourth root of radiated energy, and the fourth-root transform is used in statistics to stabilize the spread of a distribution.