Square root calculator

Roots and radicals

The square root of x is the number that, multiplied by itself, gives x: √144 = 12 because 12 × 12 = 144. A cube root works with three factors: ∛27 = 3. Numbers like 144 are perfect squares; most numbers are not, and their roots have endless decimals: √2 = 1.41421356…

In algebra and geometry, roots are usually written in simplified radical form: take out the largest perfect square. 72 = 36 × 2, so √72 = √36 × √2 = 6√2. This keeps answers exact instead of rounding.

Worked example

A square room with an area of 72 m² has sides of √72 ≈ 8.485 m. Its diagonal is the side × √2 = 6√2 × √2 = 12 m exactly.

Negative numbers

Odd roots of negative numbers are real: ∛−8 = −2. Even roots of negative numbers are not real numbers; √−4 is the complex number 2i.

The page computes the n-th root as |x| raised to 1/n, with the sign kept for odd roots of negative numbers. It checks whether the rounded result, raised back to the n-th power, gives exactly your number, in which case the answer is a whole number and the input is a perfect power. For square roots of whole numbers, it searches downward from √x for the largest square that divides x and writes the remainder under the radical. Everything runs in your browser.