Graph of the Fifth Root Function $y = \sqrt[5]{x}$

$y = \sqrt[5]{x}$ returns the number whose fifth power is $x$¹. Because $5$ is odd, a value exists for negative arguments as well, so the domain is the whole real line. Even roots, by contrast, are defined only for $x \geq 0$.

Domain and symmetry

• Both the domain and the range are all real numbers
• The function increases monotonically
• Odd function
• The tangent at the origin is vertical

A negative number has a real fifth root because multiplying a negative number five times leaves the sign negative, which is exactly what fails for a fourth root or a square root.

Sample points

Needing $x$ to reach $32$ before the value reaches $2$ is a vivid measure of how slowly it grows.

Monotonicity

Writing $y = x^{1/5}$, the derivative is $y' = \dfrac{1}{5}x^{-4/5}$. The denominator is positive whenever $x \neq 0$, so the function increases everywhere. As $x \to 0$ we get $y' \to +\infty$, so the curve rises steeply through the origin and then flattens out on both sides.

Concavity

The second derivative is $y'' = -\dfrac{4}{25}x^{-9/5}$.

The derivative is undefined at the origin, but the bending genuinely does reverse there, so it counts as an inflection point with a vertical tangent.

Inverse function

Solving for $x$ gives $x = y^5$, so the inverse is the quintic $y = x^5$, and the two graphs are reflections of each other in the line $y = x$. Where $x^5$ lies flat near the origin and shoots up far away, the fifth root stands up near the origin and lies down far away, exactly as a mirror image should.

Comparison with other roots

It meets $\sqrt[3]{x}$ at $x = 0$ and $x = \pm 1$.

At $x = 32$ we have $\sqrt[5]{32} = 2$ against $\sqrt[3]{32} \approx 3.175$. Higher roots flatten the graph further, tending in the limit toward a horizontal line at height $1$ for $x > 0$. The fourth root, meanwhile, cannot be drawn for $x < 0$ at all.

Applications

Abel and Ruffini showed that the general quintic equation cannot be solved using arithmetic and radicals², yet the simplest case, $x^5 = a$, is solved by a radical itself, and this function supplies that solution.

In physics the radius of a blast wave is given in terms of the energy, the time and the density as follows³.

Taylor famously used this one-fifth power law to deduce the yield of a nuclear test from nothing but published photographs.

1. Nth root, Wikipedia
2. Abel-Ruffini theorem, Wikipedia
3. Blast wave, Wikipedia