Nth Root Calculator — find the nth root of a number

How it works

The nth root of x is the number that, raised to the power n, gives back x — the inverse of taking a power. Written ⁿ√x or x^(1/n):

ⁿ√x = r where rⁿ = x

It unifies the roots you already know: n = 2 is the square root, n = 3 the cube root. The sign rule depends on whether n is even or odd:

• Odd n (3, 5, 7, …): defined for all numbers; the root of a negative is negative (⁵√−32 = −2).
• Even n (2, 4, 6, …): defined only for non-negative numbers, since no real number raised to an even power is negative.

Worked example

Find the 5th root of 32. Ask what number to the 5th power makes 32 — that's 2, since 2 × 2 × 2 × 2 × 2 = 32. Likewise the 4th root of 81 is 3 (3^4 = 81). Because 5 is odd, the 3rd root of −27 is −3 ((−3)³ = −27); but the 4th root of −16 has no real value, since an even power can't be negative.

Frequently asked questions

How do I calculate an nth root?

Find the number that, raised to the power n, gives x — or equivalently, raise x to the power 1/n. The 3rd root of 125 is 5 (5³ = 125). For values that aren't perfect nth powers the root is irrational, and this calculator computes it to several decimals.

What's the relationship between roots and powers?

They're inverse operations, and a root is just a fractional power: ⁿ√x = x^(1/n). Raising to 1/n undoes raising to n. This is why the nth root and the nth power cancel: (x^(1/n))^n = x.

Why can't I take an even root of a negative number?

Because raising any real number to an even power gives a non-negative result — so nothing real, squared or raised to the 4th, equals a negative. Even roots of negatives exist only among complex numbers. Odd roots are fine, because an odd power keeps the sign.

How is the nth root different from the square and cube roots?

It's the same idea for any n. The square root is the special case n = 2 and the cube root n = 3; this calculator lets you pick any n. Set n = 2 here and you get the square root; set n = 3 and you get the cube root.

What does a fractional or large n do?

A larger n pulls the root closer to 1 — higher roots of a number above 1 shrink toward 1 (the 100th root of 2 is about 1.007). This calculator takes whole-number roots; a fractional power like x^(2/3) is a power-of-a-root and belongs on the exponent calculator.

Where are nth roots used?

In geometry (the side of an n-dimensional hypercube of known volume), in finance (the nth root gives an average growth rate from n periods), and anywhere you reverse repeated multiplication — a quantity that grew by the same factor n times.