A square root undoes squaring: it asks what number, multiplied by itself, gives your value. This guide explains what a radical is, how to simplify one to its exact form, the difference between the principal and the negative root, and why some roots are rational while most are irrational.

What a radical really means

The radical sign √ asks for a root. A square root of x is a number that squares back to x; a cube root cubes back to it, and an n-th root raises to the n-th power to recover x. Algebraically a root is just a fractional exponent: √x = x^(1/2) and the n-th root is x^(1/n).

The number under the sign is the radicand, and the small number on the sign is the index. A square root leaves the index unwritten, so √ alone always means the square root.

Simplifying surds to exact form

Most roots are irrational, but many can be written more cleanly by pulling out a perfect-square factor. The rule is √(a² · b) = a√b. To simplify √72, factor it into primes — 72 = 2³ × 3² — and group each prime in pairs: one pair of 2s and one pair of 3s escape as a 2 and a 3 (giving 6), leaving a single 2 under the root. The result, 6√2, is exact and far tidier than 8.4853…

The same idea works for any index: for a cube root, pull out factors whose exponent is a multiple of three. This calculator's Simplify Radical tab shows each step so you can follow the factoring by hand.

Principal vs. negative root, and the imaginary case

Every positive number has two square roots — for example 5 and −5 both square to 25. The radical symbol returns the principal (non-negative) root, so √25 = 5, while the equation x² = 25 has both 5 and −5 as solutions. Keeping that distinction straight avoids a common algebra slip.

Negative radicands depend on the index. An even root of a negative number is imaginary — there is no real number whose even power is negative — so √(−9) = 3i. An odd root of a negative number is a real negative value, because a negative raised to an odd power stays negative: the cube root of −27 is −3.

Rational vs. irrational roots

The square root of a perfect square (1, 4, 9, 16, 25…) is a whole number and therefore rational. Every other whole number has an irrational square root: a decimal that never ends and never repeats, like √2 = 1.41421356… This is why exact radical form matters — 6√2 captures the value perfectly, while any decimal is only an approximation. The perfect-square badge in this tool tells you instantly which case you are in.