Root Calculator (Square, Cube and Nth Root)

How it works

The n-th root of a number x is the number that, multiplied by itself n times, gives x. The square root of 50 is about 7.071 because 7.071 × 7.071 ≈ 50; the cube root of 27 is exactly 3. In exponent form the n-th root is x^(1/n), which is how the calculator evaluates it.

Every positive number has one positive n-th root, and that is the one reported (the principal root). Negative numbers have a real root only when n is odd: the cube root of -8 is -2 because (-2)^3 = -8, but no real number squared gives -16, so an even root of a negative number is reported as not real.

For a whole-number radicand the calculator also writes the root in simplest radical form, the way algebra classes ask for it. It factors the number, pulls every complete n-th power outside the root sign, and leaves the rest inside: 50 = 25 × 2, so √50 = √25 × √2 = 5√2. The decimal value is unchanged; the radical form is exact.

Formula

n-th root of x = x^(1/n)        (principal root; x ≥ 0, or x < 0 with n odd)
n-th root of (-x) = -(n-th root of x)    for odd n
simplified radical: write |x| = k^n × r with k as large as possible,
then n-th root of x = k × (n-th root of r)      e.g. √50 = √(25 × 2) = 5√2

Example

√50: 50 = 25 × 2 and 25 is a perfect square, so √50 = 5√2 ≈ 7.071068. The cube root of 54 works the same way with cubes: 54 = 27 × 2, so ∛54 = 3∛2 ≈ 3.779763.

The fifth root of 32 is exactly 2, because 2^5 = 32. The cube root of -8 is -2, while the square root of -16 has no real value.

Assumptions and limitations

• Only the principal real root is shown. Every number also has complex roots (a square root of -16 is 4i), which this calculator does not report.
• Radicals are simplified only for whole-number radicands up to 1,000,000,000,000; decimals and larger numbers get the decimal root alone.
• Decimal values are cleaned to 15 significant digits and shown to 6 decimal places. Roots that are exactly whole numbers, such as ∛27 = 3, are reported exactly.

Frequently asked questions