How is the nth root defined?

The nth root is x raised to the power 1/n: the number that, raised to the nth power, returns the original x. Index 2 gives the square root and index 3 the cube root.

How does an nth root relate to raising to a power?

The nth root is the inverse operation: while a power multiplies a base by itself, the nth root of x is x to the 1/n and answers what number to the nth power gives x.

Can it find roots of negative numbers?

Only when the index n is odd; then a real root exists and is negative (for example, the cube root of −8 is −2). For a negative x with an even index, it reports that no real root exists.

Square Root and Nth Root Calculator

Find the square root, cube root, and nth root of any number at once. Supports real roots of negative numbers with odd indices, with a graph.

Last updated: June 27, 2026

Input

Enter a number x and an index n to compute the nth root, which is x to the power of 1 over n. The square root and cube root are shown as well.

Number x

The number to take the root of. Negative numbers are allowed.

Index n

The degree of the root. Use 2 for the square root and 3 for the cube root.

Result

The 2th root of x

Square root

Cube root

2.5198421

Nth root

Check (nth root to the 2th power)

When x is negative, a real root exists only when the index n is odd. For an even index the result is shown as no real root.

How it works

The nth root is defined as x raised to the power of 1 divided by n. Raising it to the nth power returns the original number x.
When the index n is 2 the result is the square root, and when n is 3 it is the cube root.
For a positive x the calculator returns the positive real root. When x is 0 the root is also 0.
When x is negative a real root exists only when the index n is odd, and that value is negative. For example, the cube root of minus 8 is minus 2.
When x is negative and the index n is even, no real root exists, so the result is shown as no real root.
The verification field shows the nth root raised back to the nth power, which in theory equals the x you entered.

Frequently asked questions

How is the nth root defined?: The nth root is x raised to the power 1/n: the number that, raised to the nth power, returns the original x. Index 2 gives the square root and index 3 the cube root.
How does an nth root relate to raising to a power?: The nth root is the inverse operation: while a power multiplies a base by itself, the nth root of x is x to the 1/n and answers what number to the nth power gives x.
Can it find roots of negative numbers?: Only when the index n is odd; then a real root exists and is negative (for example, the cube root of −8 is −2). For a negative x with an even index, it reports that no real root exists.

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