Root Calculator
Square roots, cube roots and any nth root of a number.
- Root3.000000
Inputs used
- Number27
- Root degree (n)3
About this calculator
The nth root of a number is the value that, multiplied by itself n times, gives the number back: the cube root of 27 is 3 because 3 × 3 × 3 = 27. Degree 2 is the square root, degree 3 the cube root.
Odd-degree roots work for negative numbers (the cube root of −8 is −2), but even-degree roots of negatives have no real answer — there's no real number whose square is negative — so the calculator reports an error for those.
A root is the same as a fractional exponent: the nth root of x equals x^(1/n), which is how roots slot into the rules of exponents.
Quick reference
Roots by number and degree
The nth root of each number (rows), where n is the degree in the column heading.
| Number | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| 4 | 2.000000 | 1.587401 | 1.414214 | 1.319508 |
| 8 | 2.828427 | 2.000000 | 1.681793 | 1.515717 |
| 16 | 4.000000 | 2.519842 | 2.000000 | 1.741101 |
| 27 | 5.196152 | 3.000000 | 2.279507 | 1.933182 |
| 64 | 8.000000 | 4.000000 | 2.828427 | 2.297397 |
| 81 | 9.000000 | 4.326749 | 3.000000 | 2.408225 |
| 100 | 10.000000 | 4.641589 | 3.162278 | 2.511886 |
| 125 | 11.180340 | 5.000000 | 3.343702 | 2.626528 |
| 1,000 | 31.622777 | 10.000000 | 5.623413 | 3.981072 |
Frequently asked questions
What is an nth root?
The nth root of a number is the value that, multiplied by itself n times, gives that number. The cube root of 27 is 3 because 3 × 3 × 3 = 27; degree 2 is the square root and degree 3 the cube root.
Can you take an even root of a negative number?
Not within the real numbers. No real value squared (or raised to any even power) is negative, so square and fourth roots of negatives have no real answer. Odd roots are fine — the cube root of −8 is −2.
Is a root the same as a fractional exponent?
Yes — the nth root of x equals x^(1/n). The square root of 16 is 16^(1/2) = 4, which is why roots follow the same rules as exponents.
What is the cube root of 64?
Four, because 4 × 4 × 4 = 64. Its square root is 8 and its sixth root is 2, since 2⁶ = 64.
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Results are informational only — not financial, medical or legal advice.