Root

Root

A root asks for the base. Given $b^n=x$, the $n$th root of $x$ is the value $b$ such that $b^n=x$. It is written $\sqrt[n]{x}$ or $x^{1/n}$.

Identities

$\sqrt[n]{x}\cdot \sqrt[n]{y}=\sqrt[n]{xy}$ $\frac{\sqrt[n]{x}}{\sqrt[n]{y}}=\sqrt[n]{\frac{x}{y}}(y\ne 0)$ $\sqrt[n]{x^m}=x^{m/n}$ $\sqrt[n]{\sqrt[m]{x}}=\sqrt[nm]{x}$ $\sqrt[n]{1}=1$ $\sqrt{x}=x^{1/2}$

These identities are safest over positive real numbers. Over all real numbers, even roots require non-negative radicands, and simplifications need absolute values: $\sqrt{x^2}=|x|$.

Root function

For fixed $n$,

$$f_{root}(x)=\sqrt[n]{x}=x^{1/n}.$$

It is the inverse of the power function $x\mapsto x^n$ only after choosing a suitable domain. For example, the square-root function in real calculus is usually the inverse of $x^2$ restricted to $x\ge 0$.

Calculus

Using the power rule,

$$\frac{d}{dx}\sqrt[n]{x}=\frac{d}{dx}{x}^{1/n}=\frac{1}{n}{x}^{1/n-1}.$$

For the square root,

$$\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}.$$

This derivative blows up near $x=0^{+}$, which matches the graph becoming vertical there.

Complex and postgraduate bridge

In complex analysis, roots are generally multi-valued. For example, $1$ has $n$ distinct complex $n$th roots. This is the same issue behind complex logarithm branches: writing $z^{1/n}=e^{\text{Log}(z)/n}$ requires choosing a branch of $\text{Log}$.

Common pitfalls

• Cancelling powers and roots without checking sign or domain.
• Forgetting that even roots of negative real numbers are not real.
• Treating $\sqrt{x+y}$ as $\sqrt{x}+\sqrt{y}$; this is false in general.

Numerical note

Computers often evaluate roots using powers and logarithms internally, but integer roots can be more delicate because of rounding. For exact algebra, keep expressions symbolic where possible: $\sqrt{8}=2\sqrt{2}$ is often more useful than a decimal approximation.

See also Exponentiation and Logarithm.