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Exponents, Roots & Logarithms

Nth Root Calculator

Calculate a supported integer-index real root.

Exponents, Roots & Logarithms

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  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example

Use finite decimal numbers. Calculator-specific integer and real-domain requirements are validated explicitly. Inputs stay on this device.

Result

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Understand the subject

What is Nth Root?

Exponents describe repeated multiplication, roots undo a power relationship, and logarithms name the exponent needed to reach a value.

Use it to move transparently between a base, exponent, root, logarithm, and equivalent notation.

The relationship

Write the rule before calculating

See the structure

What the calculation is doing

Worked interpretation

Read the result in context

The fourth root of 16 is 2.

Interpret with care

Important boundary

Even roots require a non-negative real radicand here, while logarithms require a positive argument and valid base.

Use the Mathematics collection to move between connected concepts without duplicating the calculation.

Extended domain

All complex nth roots

Every nonzero complex number has exactly n distinct nth roots, equally spaced around a circle in the complex plane.

Try an example
COMPLEX RESULTr1 = 1r2 = -0.5 + 0.866025403784ir3 = -0.5 − 0.866025403784i

Principal branch

ReImr1r2r3

Understand the subject

What is an nth root?

An nth root reverses raising a number to the nth power.

Use roots to recover original dimensions, undo power relationships, and solve growth or scaling equations.

The relationship

The defining equation

n is a positive whole-number index, x the radicand, and r the principal real root when n is even.

See the structure

Undo a repeated power

Worked example

Find the fourth root of 16

2⁴=16, so ⁴√16=2

Interpret with care

What the result does—and does not—mean

Even-index real roots require nonnegative radicands and return the principal nonnegative root. Odd roots can preserve the sign of negative radicands.

Quick guide

How to use this calculator

  1. Choose a calculation mode when the page offers more than one relationship.
  2. Enter the labelled known values.
  3. Read the primary result and use the check or equivalent form to verify its meaning.

Calculation method

Apply the exponent, root, or logarithm relationship

root_n(x) = x^(1/n), subject to the real domain.

The calculator checks domains, zero denominators, unsupported powers, branches, and finite-range limits before reporting a result.

Worked example

Worked example

The fourth root of 16 is 2.

root_n(x) = x^(1/n), subject to the real domain.

Supported inputs

Precision and limits

Real and complex scope

The primary panel handles the familiar real case; the extended-domain panel accepts real and imaginary parts, identifies the selected logarithm branch, and plots complex results.

Numerical range

Inputs may be zero or have absolute magnitude from 1e-100 through 1e100. Exact integer radical simplification uses its narrower disclosed bound.

Precision

Numerical functions use double-precision arithmetic and normally display 12 significant digits. Symbolic labels are exact only where explicitly stated.