Formula
ⁿ√x = x^(1/n)ⁿ√(a^n × b) = a × ⁿ√b
- x ≥ 0 for even roots
- Odd roots of negative numbers are negative
The n-th root of x is the number that gives x when raised to the power n.
Worked examples
√72
- Number
- 72
- Root index (2 = square root, 3 = cube root)
- 2
√72
8.48528137424
Simplified radical form: 6√2
Cube root of 27
- Number
- 27
- Root index (2 = square root, 3 = cube root)
- 3
∛27
3
Perfect power?: Yes — 3^3 = 27
How it works
The square root of x is the non-negative number that, multiplied by itself, gives x; the cube root is the number that gives x when multiplied by itself twice more, and the n-th root generalises this. The square root of 81 is 9 because 9 × 9 = 81.
When the number is a whole number, the calculator also pulls out any perfect-power factors to give the simplified radical form: 72 = 36 × 2, so √72 = 6√2. If nothing can be pulled out the root is irrational and cannot be simplified. A negative number has no real square root; for the square root the tool shows the imaginary result i·√|x|, while an odd root of a negative number is simply negative.
For powers use the Exponent Calculator, and for the factors behind a simplified radical see the Prime Factorization Calculator. To use a root inside a longer expression try the Scientific Calculator.
Assumptions and limits
- Real roots, except that the square root of a negative number is shown as an imaginary value.
- Simplified radical form is shown for whole numbers up to 10¹².