Square Root Calculator (and Cube and Nth Roots)

This root calculator finds square roots, cube roots and any nth root, accurate to 12 significant figures. For whole numbers it also writes the answer in simplest radical form — the form math classes ask for, such as √72 = 6√2 — and tells you whether the number is a perfect square, cube or higher power. Negative numbers work too: odd roots are real, and even roots are given as imaginary or complex numbers.

How to use this tool

1. Enter the number.
2. Enter the root index: 2 for a square root, 3 for a cube root, or any whole number up to 100.
3. Read the decimal value, the simplified radical and the perfect-power check.
4. The table lists the 2nd to 10th roots of the same number.

Formula

ⁿ√x = x^(1/n) · Simplify: ⁿ√(aⁿ × b) = a ⁿ√b

To simplify, the number is split into prime factors and every group of n equal factors moves outside the root: 72 = 2³ × 3² = (2² × 3²) × 2, so √72 = 2 × 3 × √2 = 6√2. Odd roots of negative numbers are negative (∛−8 = −2); square roots of negatives are imaginary (√−4 = 2i).

Example

The square root of 72 is about 8.48528137424. Since 72 = 36 × 2 and 36 is a perfect square, √72 = 6√2. The cube root of 54 is 3∛2 ≈ 3.77976314968, because 54 = 27 × 2.