Square Root Calculator
Find the square root or any nth root of a number, with the simplified exact radical form, a perfect-power check, and the negative root for even roots.
Como usar esta calculadora
- 1Enter the number you want the root of.
- 2Set the root: 2 for a square root, 3 for a cube root, and so on.
- 3Read the principal root and, for even roots, the negative one too.
- 4Check whether it's a perfect power or an irrational number, and see the simplified radical.
Como funciona
Roots
√x is the number that, squared, gives x ⁿ√x = x^(1/n) — the number that, to the nth power, gives x every positive x has roots +√x and −√x for even n odd roots exist for negatives; even roots of negatives are imaginary
A root is the inverse of a power. The square root of x is the number that, multiplied by itself, gives x; the cube root is the number that, cubed, gives x; and in general the nth root is x raised to the power 1/n. For even roots there are two real answers of opposite sign, because both a number and its negative give the same even power — the principal root, written with the radical sign, is the non-negative one. Odd roots behave more simply: there is exactly one real root, and it can be negative, since raising a negative number to an odd power keeps it negative. The awkward case is an even root of a negative number, which has no real value at all and lives in the imaginary numbers, because no real number squared can be negative.
Exemplo resolvido
The square root of 144 is 12, because 12 × 12 = 144 — and −12 is equally a square root, since (−12)² is also 144. The cube root of −27 is −3, a real answer, because (−3)³ = −27. But the square root of −4 has no real value; it is the imaginary number 2i.
Square Root Calculator: o guia completo
Why every positive number has two square roots
It surprises many people that a positive number has not one square root but two. The square root of 25 is usually given as 5, yet −5 is just as valid, because (−5) × (−5) = 25 as well. Both 5 and −5, when squared, land on 25. The number written after the radical sign is called the principal square root, and by convention it is always the non-negative one, which is why √25 = 5 and not ±5. But when you solve an equation like x² = 25, you must keep both roots, giving x = ±5.
This distinction between the principal root and the full set of roots is a frequent source of algebra mistakes. Taking the square root of both sides of an equation introduces the ± that the radical symbol alone suppresses. The calculator shows the negative root explicitly for even roots precisely to keep this visible. For odd roots the issue disappears: the cube root of 8 is simply 2, with no second real answer, because cubing preserves sign.
Perfect powers, irrationals, and exact form
Some numbers have roots that come out as tidy whole numbers — 144 is a perfect square with root 12, and 27 is a perfect cube with root 3. Most numbers do not. The square root of 2, famously, is irrational: its decimal expansion never ends or repeats, and it cannot be written as a fraction. This was a genuine shock to ancient mathematicians, who had assumed every length could be expressed as a ratio of whole numbers; the diagonal of a unit square proved otherwise.
When a square root is irrational, it can still often be simplified into an exact radical form rather than a rounded decimal. The square root of 72, for instance, is not a whole number, but 72 = 36 × 2, so √72 = 6√2 — a form that is both exact and more compact than a string of decimals. Pulling out the largest perfect-square factor is the standard technique, and it keeps answers precise for further algebra, where a rounded decimal would introduce error. The calculator shows this simplified radical for square roots whenever one exists.
Roots, fractional exponents, and imaginary numbers
Roots and powers are two views of the same operation. Writing the nth root as the fractional exponent 1/n makes the connection explicit and lets the ordinary laws of exponents handle roots: the fourth root of a number is the same as raising it to the power 0.25, and combined roots and powers like x^(2/3) mean 'cube root, then square'. This unification is why calculators compute roots through exponentiation, and why understanding one deepens the other.
The boundary of the real numbers appears at the even root of a negative. Because no real number multiplied by itself yields a negative result, √−1 cannot be real — so mathematicians defined it as i, the imaginary unit, opening the door to the complex numbers. Far from being a mere curiosity, this extension turned out to be essential to physics and engineering, underpinning the description of alternating current, quantum mechanics, and wave behaviour. The 'no real root' this calculator reports for an even root of a negative is not a dead end but the entrance to that larger number system.
Perguntas frequentes
How do I calculate a square root?
The square root of a number is the value that, multiplied by itself, gives that number — √144 = 12 because 12 × 12 = 144. For non-perfect squares the root is irrational, so a calculator gives a decimal approximation, though it can often be simplified into exact radical form.
Does a number have two square roots?
Every positive number has two square roots, one positive and one negative, since both give the same square — 5 and −5 both square to 25. The radical sign √ denotes the principal (non-negative) root, but solving x² = 25 keeps both, giving x = ±5.
Can you take the root of a negative number?
Odd roots yes — the cube root of −27 is −3, a real number, because a negative cubed stays negative. Even roots of negatives have no real value; the square root of −4 is the imaginary number 2i. This is the boundary where the real numbers give way to the complex numbers.
How do you simplify a square root?
Factor out the largest perfect square. Since 72 = 36 × 2 and √36 = 6, √72 simplifies to 6√2. This exact radical form is more precise than a rounded decimal and is preferred in algebra, where rounding would introduce error into later steps.