Une racine carrée défait la quadrature : elle demande quel nombre, multiplié par lui-même, donne votre valeur. Ce guide explique ce qu'est un radical, comment le simplifier jusqu'à sa forme exacte, la différence entre la racine principale et la racine négative et pourquoi certaines racines sont rationnelles alors que la plupart sont irrationnelles.
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 radican, and the small number on the sign is the indice. 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.
Racine principale ou racine négative et cas imaginaire
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 valeur, car un négatif élevé à une puissance impaire reste négatif : la racine cubique de −27 est −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 capture parfaitement la valeur, alors que toute décimale n'est qu'une approximation. Le badge carré parfait de cet outil vous indique instantanément dans quel cas vous vous trouvez.