Root Calculator

Square roots, cube roots and nth roots with simplified radical form, plus the complex roots of negative numbers.

Your result will appear here

Fill in the fields and press Calculate, or pick one of the examples.

About the Root Calculator

Finds square roots, cube roots and any nth root. For whole-number radicands the root is also written in simplified radical form, for example $\sqrt{72} = 6\sqrt{2}$ .

Odd roots of negative numbers are real ( $\sqrt[3]{-27} = -3$ ). Even roots of negative numbers have no real value, so the principal complex root is given, together with the full list of complex roots.

Limitations

Radical simplification is offered for whole-number radicands up to 10¹⁵ and indices up to 12.

How to use it

Enter the number under the root.
Enter the index: 2 for a square root, 3 for a cube root, and so on.
Press Calculate.

Example

√72

→ 6√2 ≈ 8.48528137424

Formulas used

\sqrt[n]{a^{n} b} = a\sqrt[n]{b}

Simplify a radical by pulling out perfect nth-power factors.

z_k = \sqrt[n]{r}\, e^{i(\theta + 2\pi k)/n},\quad k = 0, \dots, n-1

Every non-zero number has exactly n complex nth roots.

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About the Root Calculator

Limitations

How to use it

Formulas used

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About the Root Calculator

Limitations

How to use it

Formulas used