Root calculator

Square, cube and nth roots, with exact simplified surds where possible.

Formula Maths

Root

Rounded for display

Simplified radical: 6√2

How it works

The nth root of x is the number that gives x when multiplied by itself n times. Square roots and other even roots of positive numbers have a positive principal value.

To simplify a radical, the calculator takes out every factor that appears n times: 72 = 6² × 2, so √72 = 6√2.

Odd roots of negative numbers are negative and real. Even roots of negative numbers are complex: √−16 = 4i.

Formula

ⁿ√x = x^(1/n)
√(a² × b) = a√b
√(−x) = i√x

Worked example

√72 = √(36 × 2) = 6√2 ≈ 8.485281374. ∛27 = 3 exactly. ∛−8 = −2, while √−16 = 4i.

Questions people ask

Does a square root have two answers?

The equation x² = 16 has two solutions, 4 and −4. The square root symbol means the positive one, 4.

How do I simplify a square root?

Find the largest square factor, take its root outside, and leave the rest inside: √50 = √(25 × 2) = 5√2.

What is i?

i is the imaginary unit, defined by i² = −1. It lets even roots of negative numbers be written: √−9 = 3i.