Calculate square roots, cube roots, and n-th roots with simplified radical form, prime factorization, and precise decimal values.
Find the prime factors of 72 to identify any perfect 2-th power factors.
Pull out 6 from under the radical sign.
The n-th root of a real number x (denoted ⁿ√x) is a number r such that rⁿ = x. While every positive number has both a positive and negative square root (e.g. √9 = ±3), the radical symbol √ specifically designates the principal (non-negative) root. Our calculator provides exact simplified radical expressions (k ⁿ√r), decimal approximations, and prime factor extractions.
Factor the radicand into primes. Any prime factor with power p ≥ n can have factor ⌊p/n⌋ moved outside the radical.
Odd roots of negative numbers are real (e.g. ³√(-8) = -2). Even roots of negative numbers require complex numbers (e.g. √(-4) = 2i).
Roots of the same degree can be combined: ⁿ√a · ⁿ√b = ⁿ√(ab), and ⁿ√a / ⁿ√b = ⁿ√(a/b).
Fractions with radicals in the denominator are conventionally multiplied by ⁿ√(xⁿ⁻¹) to leave an integer denominator.
Simplify the square root √72 into simplest radical form k√r and evaluate its decimal value.
Because (-4)² = +16, not -16. There is no real number that squares to -16. Mathematicians define the imaginary unit i = √(-1), giving √(-16) = √(16 · (-1)) = 4i.
Yes! Because (-2)³ = (-2) × (-2) × (-2) = -8, the cube root ³√(-8) = -2 is a completely valid real number.
While both +3 and -3 squared equal 9, by universal convention the radical symbol √x denotes only the principal (non-negative) root, so √9 = 3.
A root is simply an exponent with a unit fraction: the square root is x^(1/2), the cube root is x^(1/3), and the n-th root is x^(1/n).
Understanding your calculation
The nth root of x is a value whose nth power equals x. A square-root result conventionally denotes the nonnegative principal root.
The cube root of 27 is 3, and the square root of 81 is 9.
Even roots of negative values are not real numbers. Odd roots can be negative; numerical approximations and unsupported complex results need careful interpretation.