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Walk-through

How to Use This Calculator

3 steps

1

Enter the number and root index

On the Nth Root tab, type the number you want the root of (the radicand) and the root index n — 2 for a square root, 3 for a cube root, or any whole number 2 or higher. The result updates instantly as you type.

2

Read the result

The result card shows the principal (positive) root as a decimal. If you entered a negative number with an even index, the calculator flags that there is no real result and explains why — the root is imaginary.

3

Try Square/Cube or Power mode

Switch to the Square/Cube tab to see a number's square root and cube root at once, or the Power tab to run the inverse operation — raising a base to an exponent (x^n) — including fractional and negative exponents.

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Reference

Formula & Methodology

2 formulas

Nth root

root = x^(1/n)

The nth root of x is the number that, multiplied by itself n times, gives x. It is computed as x raised to the power 1/n. When x is negative and n is odd, the result is a real negative number; when n is even, there is no real result (the answer is imaginary).

Power (inverse)

result = x^n

Raising x to the power n is the inverse of taking an nth root — it 'undoes' the root operation. The exponent n can be a whole number, a fraction, or negative. A negative base raised to a non-integer exponent has no real result.

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Glossary

Key Terms Explained

6 terms

Root A value that, when raised to a given power (the index), produces the original number. The square root of 9 is 3 because 3^2 = 9.

Radical The root symbol (√) and the expression it applies to, such as √16 or ∛27. 'Radical form' means expressing a root using this symbol rather than as a decimal.

Index The small number written to the upper-left of the radical symbol (the n in ⁿ√x) that tells you which root to take — 2 for square root, 3 for cube root, and so on. When no index is shown, it is assumed to be 2.

Radicand The number underneath the radical symbol — the value you are taking the root of. In √16, the radicand is 16.

Principal root The positive real root of a number, by mathematical convention. Even though (-3)^2 = 9 just like 3^2 = 9, the principal square root of 9 is defined as +3, not -3.

Fractional exponent An exponent written as a fraction, like x^(1/3), which represents the same operation as taking a root — x^(1/3) is identical to ∛x.

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Scenarios

Real-World Examples

3 worked examples

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Cube root check

Nth Root, index 3

Number 27 Root index 3

27^(1/3) = 3, since 3 × 3 × 3 = 27. This is an exact whole-number cube root.

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Irrational square root

Nth Root, index 2

Number 2 Root index 2

√2 ≈ 1.4142 (rounded to 4 decimal places). Unlike √4 or √9, this root never terminates or repeats — it is irrational.

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4th root

Nth Root, index 4

Number 16 Root index 4

The 4th root of 16 is 2, because 2 × 2 × 2 × 2 = 16. Switch to Power mode with base 2 and exponent 4 to confirm the inverse.

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Deep Dive

Understanding the Root Calculator

3 chapters · 4 min read

A root is the inverse of raising a number to a power: while 3^2 = 9 tells you what 3 squared equals, a square root asks the reverse question — what number, squared, gives 9? This calculator finds the answer for any root index, plus its inverse operation (raising a number to a power), and explains the tricky negative-number cases along the way.

How the Root Calculator works

The calculator computes the principal nth root of a number using the identity x^(1/n) — the same operation as a fractional exponent. For the common cases (square root, cube root) it uses the equivalent of a computer's built-in square-root and cube-root functions for maximum precision; for any other index it raises the number to the power 1/n directly.

The trickiest part is sign handling. A positive number always has a well-defined real root. A negative number's root depends on whether the index is even or odd: an odd index (3, 5, 7, …) gives a real negative result, because a negative number raised to an odd power stays negative. An even index (2, 4, 6, …) has no real result, because no real number multiplied by itself an even number of times can produce a negative value — the true answer involves imaginary numbers, which this calculator flags rather than silently returning a wrong number.

Inputs and what they mean

On the Nth Root tab, Number is the radicand — the value you're taking the root of, positive or negative. Root index is a whole number of 2 or greater that says which root to take (2 = square root, 3 = cube root, 4 = fourth root, and so on).

On the Power tab, Base is the number being raised to a power, and Exponent is the power itself — this can be a whole number, a fraction, or negative, since raising to a power is the direct inverse of taking a root.

Limits and edge cases

This calculator reports the principal (positive real) root only — it does not enumerate every complex root of a number, which is a topic for a full complex-analysis tool. When an even-index root of a negative number is requested, the result is flagged as having no real value rather than displaying an incorrect number.

Very large exponents or radicands can exceed the precision of standard floating-point arithmetic; results are rounded to a reasonable number of decimal places and should be treated as approximations for irrational values like √2 or √3.

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Questions

Frequently Asked Questions

6 questions

What is the formula for a root?+

The nth root of a number x is x raised to the power 1/n, written x^(1/n). For a square root, n = 2, so √x = x^(1/2). For a cube root, n = 3, so ∛x = x^(1/3).

Can you take the cube root of a negative number?+

Yes. Because the index (3) is odd, a negative radicand has a real cube root — for example, the cube root of -8 is -2, since -2 × -2 × -2 = -8. Any odd-index root of a negative number works the same way.

What happens with the square root of a negative number?+

There is no real square root of a negative number, because squaring any real number (positive or negative) always gives a positive result. This calculator flags that case as imaginary rather than returning an incorrect decimal — the true answer involves i, the imaginary unit, where i² = -1.

What is radical form?+

Radical form expresses a root using the radical symbol (√, ∛, etc.) rather than as a decimal approximation — for example, writing √2 instead of 1.4142. This calculator shows the decimal value alongside a note of the radical notation used.

What is a fractional exponent?+

A fractional exponent like x^(1/3) represents exactly the same operation as taking a root — x^(1/3) equals the cube root of x. The Power tab lets you compute any exponent, including fractional ones, to see this relationship directly.

What is the principal root?+

The principal root is the positive real root of a number, chosen by mathematical convention. For example, both 4 and -4 squared equal 16, but the principal square root of 16 is defined as +4, not -4 — this calculator always returns the principal root.

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Root Calculator

Inputs

Result

How to Use This Calculator

Enter the number and root index

Read the result

Try Square/Cube or Power mode

Formula & Methodology

Key Terms Explained

Real-World Examples

Cube root check

Irrational square root

4th root

Understanding the Root Calculator

How the Root Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Root Calculator

Inputs

Result

How to Use This Calculator

Enter the number and root index

Read the result

Try Square/Cube or Power mode

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Root Calculator

How the Root Calculator works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Keep Exploring

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