Root Calculator — Free Online Calculator with Step-by-Step Guide

How Root Calculator Works

ⁿ√x = x^(1/n)

The nth root of x is the number that, when multiplied by itself n times, equals x. It is equivalent to raising x to the power of 1/n. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27. The square root (n = 2) is the most common, but any positive integer root can be computed.

n

Root Index — The degree of the root (2 = square root, 3 = cube root, 4 = fourth root, etc.). Must be a positive integer. Even-indexed roots of negative numbers are not real.

x

Radicand — The number under the radical sign. For even-indexed roots, x must be non-negative. For odd-indexed roots, x can be negative.

ⁿ√x

Nth Root — The radical expression. The small number n is the index; x is the radicand. For square roots, the index 2 is usually omitted.

How to Use

1. Enter the root index (n) — use 2 for square root, 3 for cube root, 4 for fourth root, etc.
2. Enter the radicand (x) — the number under the radical sign.
3. View the result as both a decimal value and a fractional exponent (x^(1/n)).
4. For integers, the calculator may show a simplified radical form when the radicand has a perfect nth power factor.

Understanding the Result

The nth root of a number x is written as ⁿ√x. It is the inverse operation of exponentiation: if y = ⁿ√x, then yⁿ = x. The nth root can also be expressed as a fractional exponent: x^(1/n). For example, 27^(1/3) = ³√27 = 3 because 3³ = 27. For even indices (n = 2, 4, 6...), the radicand must be non-negative because no real number multiplied by itself an even number of times produces a negative result. For odd indices (n = 3, 5, 7...), negative radicands are allowed: ³√(-8) = -2 because (-2)³ = -8. The calculator attempts to simplify radicals by factoring out perfect nth powers. For example, √12 = 2√3 because 12 = 4 × 3 and √4 = 2. This simplified radical form is often preferred in mathematics because it is exact, unlike the decimal approximation. Roots appear throughout geometry (side lengths, diagonal calculations), physics (wave functions, inverse-square laws), and finance (calculating interest rates from growth factors).

Frequently Asked Questions

What is the difference between square root and nth root?

The square root (n = 2) finds a number that, when multiplied by itself once (two factors), equals x. The cube root (n = 3) finds a number that, when multiplied by itself twice (three factors total), equals x. The nth root extends this concept to any positive integer n. As n increases, the nth root of a number approaches 1 (for numbers greater than 1) or the number itself (for numbers between 0 and 1).

Can I take the nth root of a negative number?

Yes, when the index n is odd. For example, ³√(-8) = -2 because (-2)³ = -8. But when n is even, the result is not a real number — there is no real number that, when multiplied by itself an even number of times, gives a negative product. The calculator will return empty for even-indexed roots of negative numbers.

What does x^(1/n) mean?

x^(1/n) is the fractional exponent form of the nth root. Raising x to the power of 1/n is mathematically equivalent to taking the nth root of x. For example, 27^(1/3) = ³√27 = 3. More generally, x^(m/n) = (ⁿ√x)ᵐ = ⁿ√(xᵐ). This fractional exponent notation is often more convenient for algebraic manipulation.

Sources

• Wolfram MathWorld - nth Root
• Wikipedia - nth Root