Can I take the root of a negative number?

Only when the index is odd. The cube root of −8 is −2 because (−2) × (−2) × (−2) = −8. An even root of a negative number (like √−4) has no real value, so the calculator returns no result in that case.

What is the difference between a square root and a cube root?

A square root (index 2) asks which number multiplied by itself gives the radicand, while a cube root (index 3) asks which number multiplied by itself three times does. The same engine handles any whole or fractional index through the exponent 1/n.

Why does this return only the principal root?

Every positive number technically has two square roots (for example 3 and −3 both square to 9), but the calculator reports the principal, non-negative root. For odd indices the real root is unique, so only one value is shown.

Can the index be a decimal?

Yes. Because the result is radicand^(1/n), any nonzero n works, including fractional indices. An index of exactly 0 is undefined (it would divide by zero in the exponent) and is rejected.

Root Calculator

Radicand

Index

Result5.000000

A root is the inverse of raising a number to a power: the nth root of a value is the number that, multiplied by itself n times, returns that value. This calculator finds the square root, cube root, or any nth root of a radicand by evaluating it as a fractional exponent. It handles negative radicands for odd roots, where a real answer exists.

Formula

root = radicand^(1 / n) (for radicand < 0 and odd n: root = −|radicand|^(1 / n))

radicand: The number you are taking the root of (under the radical)
n: The root index (2 = square root, 3 = cube root); must not be 0
root: The principal nth root of the radicand

How it works

Enter the radicand (the number under the radical sign) and the root index n — for example index 2 for a square root or 3 for a cube root.
The calculator computes radicand^(1/n). For a negative radicand with an odd index it takes the positive root of the absolute value and negates it; a negative radicand with an even index has no real result and returns nothing.
Read the principal root in the result. An index of 0 is undefined and is rejected.

Worked examples

Find the cube root of 27.

Index n = 3, radicand = 27.
Compute 27^(1/3).
3 × 3 × 3 = 27, so the cube root is exactly 3.

Cube root of 27 = 3

Find the square root of 2.

Index n = 2, radicand = 2.
Compute 2^(1/2).
The result is an irrational number.

√2 ≈ 1.4142135623730951

Frequently asked questions

Can I take the root of a negative number?: Only when the index is odd. The cube root of −8 is −2 because (−2) × (−2) × (−2) = −8. An even root of a negative number (like √−4) has no real value, so the calculator returns no result in that case.
What is the difference between a square root and a cube root?: A square root (index 2) asks which number multiplied by itself gives the radicand, while a cube root (index 3) asks which number multiplied by itself three times does. The same engine handles any whole or fractional index through the exponent 1/n.
Why does this return only the principal root?: Every positive number technically has two square roots (for example 3 and −3 both square to 9), but the calculator reports the principal, non-negative root. For odd indices the real root is unique, so only one value is shown.
Can the index be a decimal?: Yes. Because the result is radicand^(1/n), any nonzero n works, including fractional indices. An index of exactly 0 is undefined (it would divide by zero in the exponent) and is rejected.

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