A root reverses a power. The principal square root $\sqrt a$ is the nonnegative real number whose square is $a$, when such a real number exists. For example,

$$\sqrt{9}=3$$

because $3^2=9$.

Higher roots

The $n$th root of a number reverses raising to the power $n$:

$$\sqrt[n]{a}=b\qquad\Longleftrightarrow\qquad b^n=a,$$

with the usual real-number restrictions. Even roots of negative real numbers are not real, while odd roots can be negative; for example,

$$\sqrt[3]{-8}=-2.$$

Rational exponents

Roots can be written as fractional powers:

$$a^{1/n}=\sqrt[n]{a}.$$

More generally,

$$a^{m/n}=\sqrt[n]{a^m}$$

whenever the expression is defined over the real numbers. This connects the exponent laws with roots.

Simplifying radicals

Perfect powers can be extracted from a radical. For example,

$$\sqrt{12}=\sqrt{4\cdot3}=2\sqrt3.$$

Radical expressions must always be interpreted together with their real domain. In particular, $\sqrt{-1}$ is not a real number.