Nth root

Calculator for computing the nth root

Nth root calculator

What is calculated?

This function returns the nth root ⁿ√x of the radicand x with the specified root exponent n. The radicand must be a non-negative real number for even n.

Input values



Result
The result is shown with the selected number of decimal places

Nth root info

Properties

Nth root:

  • Generalization of the square root
  • Inverse function of x^n
  • Defined for x ≥ 0 when n is even
  • Defined for all x when n is odd

Note: For even root exponents the radicand must be non-negative. For odd exponents negative radicands are allowed.

Examples
³√8 = 2
2³ = 8 (cube root)
⁴√16 = 2
2⁴ = 16 (fourth root)
⁵√32 = 2
2⁵ = 32 (fifth root)
⁶√64 = 2
2⁶ = 64 (sixth root)
Special cases
  • n = 2: Square root √x
  • n = 3: Cube root ³√x
  • Even n: x ≥ 0 required
  • Odd n: all real x allowed

Formulas of the nth root

Definition
\[\sqrt[n]{x} = x^{1/n}\] Nth root as a power
Inverse relation
\[\sqrt[n]{x} = y \Leftrightarrow y^n = x\] Definition via nth power
Product rule
\[\sqrt[n]{x \cdot y} = \sqrt[n]{x} \cdot \sqrt[n]{y}\] Root of a product
Quotient rule
\[\sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}\] Root of a quotient
Power rule
\[\sqrt[n]{x^m} = x^{m/n}\] Root of a power
Nested roots
\[\sqrt[m]{\sqrt[n]{x}} = \sqrt[mn]{x}\] Composite roots

Calculation example

Example: calculate ³√64

Given:

  • Radicand x = 64
  • Root exponent n = 3
  • Sought: ³√64

Calculation:

\[\sqrt[3]{64} = 4\] \[\text{since } 4^3 = 64\]

Result: The cube root of 64 is 4.

Powers of 2

Example: different roots of powers of two

Powers of 2:

  • 2¹ = 2
  • 2² = 4
  • 2³ = 8
  • 2⁴ = 16
  • 2⁵ = 32
  • 2⁶ = 64

Corresponding roots:

  • ¹√2 = 2
  • ²√4 = 2
  • ³√8 = 2
  • ⁴√16 = 2
  • ⁵√32 = 2
  • ⁶√64 = 2

Pattern: For 2ⁿ the nth root is always 2.

Geometric example

Example: determine cube edge length

Problem:

A cube has volume 125 cm³. How long is one edge?

Solution:

\[\text{Volume} = a^3 = 125\] \[a = \sqrt[3]{125} = 5 \text{ cm}\]

Application: The cube root is used to determine edge lengths from volumes.

Definition and properties

General definition

The nth root of a number x is the number y which, when multiplied by itself n times, gives x. Mathematically: yⁿ = x, so y = ⁿ√x.

Parity distinction

For even n (2, 4, 6, ...) x must be ≥ 0, since even powers of negative numbers are positive. For odd n (1, 3, 5, ...) negative values for x are allowed.

Mathematical properties
  • Even n: Domain x ≥ 0
  • Odd n: Domain ℝ
  • Monotonicity: strictly increasing
  • Continuity: continuous on the domain

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