Radical

The symbol $\root n\of x$ used to indicate a root is called a radical. The expression $\root n\of x$ is therefore read ``$x$ radical $n$,'' or ``the $n$th Root of $x$.'' $n=2$ is written $\sqrt{x}$ and is called the Square Root of $x$. $n=3$ corresponds to the Cube Root. The quantity under the root is called the Radicand.

Some interesting radical identities are due to Ramanujan, and include the equivalent forms

$$(2^{1/3}+1)(2^{1/3}-1)^{1/3}=3^{1/3}$$

and

$$(2^{1/3}-1)^{1/3}=({\textstyle{1\over 9}})^{1/3}-({\textstyle{2\over 9}})^{1/3}+({\textstyle{4\over 9}})^{1/3}.$$

Another such identity is

$$(5^{1/3}-4^{1/3})^{1/2}={\textstyle{1\over 3}}(2^{1/3}+20^{1/3}-25^{1/3}).$$

See also Cube Root, Nested Radical, Power, Radical Integer, Radicand, Root (Radical), Square Root, Surd, Vinculum