Root Calculator
Calculate any nth root — square roots, cube roots, and beyond. Shows both radical form and fractional exponent equivalent with step-by-step verification.
Nth Root
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The Formula
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.
Variable Definitions
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.
Radicand
The number under the radical sign. For even-indexed roots, x must be non-negative. For odd-indexed roots, x can be negative.
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 This Calculator
- 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.
The nth root of x answers: what number raised to the nth power equals x?
Understanding the Concept
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
Sources & References
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