Radical

A radical is the symbol used for square root or nth root. The symbol used to denote a radical is ${\sqrt[{n}]{m}}$, n is the index of the radical (n > 0) and m is called the "radicand". A radical without an index is typically a square root. Radicals and exponents are inverse operations, raising a number to the nth power and taking the nth root undo each other.

Raising a number to the power of 1/n means taking the nth root of that number, so $m^{\frac {1}{n}}={\sqrt[{n}]{m}},n>0$

If the index of a root is even and the radicand is negative, then the resulting value of the expression is a complex number.

Properties

For all of these, assume l > 0 or m > 0.

• ${\sqrt[{n}]{lm}}={\sqrt[{n}]{l}}\cdot {\sqrt[{n}]{m}}$
• ${\sqrt[{n}]{\frac {l}{m}}}={\frac {\sqrt[{n}]{l}}{\sqrt[{n}]{m}}},m\neq 0$
• $({\sqrt[{n}]{m}})^{l}={\sqrt[{n}]{m^{l}}}$
• ${\sqrt[{n}]{m^{l}}}=m^{\frac {l}{n}}$
• If n is even, then $({\sqrt[{n}]{m}})^{n}=|m|$
• If n is odd, then $({\sqrt[{n}]{m}})^{n}=m$