The nth root of x is the number that, multiplied by itself n times, gives x — written as x^(1/n). The 4th root of 81 is 3 because 3 x 3 x 3 x 3 = 81.

Can I take the root of a negative number?

Odd roots of negatives are real: the cube root of -27 is -3. Even roots of negative numbers (square, 4th, etc.) have no real solution and require imaginary numbers.

How do roots relate to fractional exponents?

A root is just a fractional exponent: x^(m/n) equals the nth root of x raised to the m. For example, 8^(2/3) is the cube root of 8 (which is 2) squared, giving 4.

How do I simplify a radical?

Factor out perfect powers matching the root index. For square roots, √72 = √(36 x 2) = 6√2; for cube roots, ∛54 = ∛(27 x 2) = 3∛2.

How can I estimate a root without a calculator?

Bracket the answer between perfect powers and interpolate: ∛50 lies between ∛27 = 3 and ∛64 = 4, closer to 3.7. Repeated averaging (Newton's method) refines the estimate quickly.

Root Calculator

Calculate square roots, cube roots, and nth roots of numbers. Free, fast, accurate — no signup, mobile-friendly, works i

Root Type

Parameters

Number

The number to find the root of

Root Index (n)

The index of the root (2 for square root, 3 for cube root, etc.)

📐 Common Roots

Root Rules

√(ab) = √a × √b

√(a/b) = √a / √b

(√a)² = a (for a ≥ 0)

√a² = |a|

ⁿ√aᵐ = a^(m/n)

Primary Result

Root Value

Perfect root

Type

Perfect

Rational

Scientific

4.0000e+0

notation

Squared

value²

Cubed

value³

Verification: Correct

Check: 64

Step-by-Step Solution

1Given: 3√64

2Formula: x = 64^(1/3)

3Principal root: 4.000000

4Verification: 4.0000^3 = 64.0000

How it works

A root calculator finds the nth root of a number — the value that, raised to the nth power, gives back the number. The square root (n=2) and cube root (n=3) are the most common; an nth root is the same as raising to the power 1/n.

nth root

ⁿ√x = x^(1/n)   such that   (ⁿ√x)ⁿ = x

x: the number (radicand)
n: the root index (2 = square, 3 = cube)

Worked example

Find the cube root of 64

Which y gives y³ = 64?
4 × 4 × 4 = 64

³√64 = 4.

Good to know

Odd roots of negative numbers are real (³√−8 = −2); even roots of negatives are not.
A fractional exponent and a root are the same thing: x^(1/3) = ³√x.
Most roots are irrational and can only be approximated as decimals.