Root Calculator
Enter a number and a root degree to compute the nth root. Square root (n=2), cube root (n=3), or any higher integer degree. Odd roots of negative numbers return a negative real result; even roots of negative numbers return a complex-number message.
³√27
3
- Radicand (x)
- 27
- Degree (n)
- 3
- Result (x^(1/3))
- 3
- Perfect root?
- Yes — integer result
The nth root of x is the number y such that yⁿ = x, computed as x^(1/n). Even roots of negative numbers have no real value; odd roots of negative numbers return a negative real result. Results are computed with IEEE-754 double-precision arithmetic.
How to use this calculator
Type the number you want to take the root of into the first field, and the root degree into the second. For a square root use degree 2, for a cube root use 3, for a fourth root use 4, and so on. Negative numbers work for odd degrees (³√−27 = −3); even degrees of negatives are rejected because the answer is not a real number. Results update as you type and switch to scientific notation when very large or very small.
How the calculation works
The nth root of x is the number y such that yⁿ = x. The calculator evaluates this as x^(1/n) using IEEE-754 double-precision arithmetic, the same standard used by spreadsheets and scientific calculators. For negative bases with odd integer degrees, the calculator computes −(|x|^(1/n)) directly to sidestep the floating-point glitch where Math.pow(−8, 1/3) returns NaN because 1/3 cannot be represented exactly. When the result is an integer (e.g. √25 = 5, ³√27 = 3), the calculator labels it a "perfect root"; otherwise the result is irrational and only its decimal approximation is shown.
Worked example
√25 = 5 because 5 × 5 = 25 (perfect square). √2 ≈ 1.41421356 — irrational. ³√27 = 3 because 3³ = 27. ³√−8 = −2 because (−2)³ = −8 (odd-degree roots of negatives are real). ⁴√16 = 2 because 2⁴ = 16. ⁵√32 = 2 because 2⁵ = 32. √−4 is rejected — the answer 2i is imaginary, not a real number.
Frequently asked questions
What is the difference between a square root, cube root, and nth root?
A square root (n=2) is the number that, multiplied by itself once, gives the original: √9 = 3 because 3×3 = 9. A cube root (n=3) is the number multiplied by itself twice: ³√8 = 2 because 2×2×2 = 8. An nth root generalises the pattern: y is the nth root of x when yⁿ = x. The degree n in this calculator can be any positive or negative integer except zero.
Why can you take the cube root of a negative number but not the square root?
Because (−2)³ = −8 — multiplying three negatives gives a negative, so −2 is a genuine real cube root of −8. For a square root, you would need a real number y with y² = −4; but any real y squared is non-negative, so no real solution exists. The general rule: odd-degree roots of negative numbers are real, even-degree roots of negative numbers are complex (involve the imaginary unit i). This calculator returns real results only and rejects even roots of negatives.
Why does the calculator say "perfect root"?
A perfect nth root is one where the result is an integer: √25 = 5, ³√27 = 3, ⁴√16 = 2. These are exact. When the root is not perfect — √2 ≈ 1.41421356, √10 ≈ 3.16227766 — the result is irrational, meaning its decimal expansion is non-terminating and non-repeating. The calculator shows the closest double-precision approximation, accurate to about 15–17 significant digits.
What does a fractional or decimal radicand do?
It works the same way. √0.25 = 0.5 because 0.5² = 0.25. ³√0.001 = 0.1 because 0.1³ = 0.001. √1.44 = 1.2 because 1.2² = 1.44. Fractional inputs are valid for any degree the calculator accepts; the rule about even roots needing a non-negative radicand still applies.
What is the relationship between roots and exponents?
They are inverses. The nth root of x equals x raised to the power 1/n: ³√27 = 27^(1/3) = 3. More generally, x^(p/q) is the qth root of x raised to the power p: 8^(2/3) = (³√8)² = 2² = 4. This is why fractional exponents and roots are the same operation written in different notation, and why a Root Calculator and an Exponent Calculator solve the same family of problems from opposite directions.
How accurate is the result?
Calculations use IEEE-754 double-precision floats, accurate to about 15–17 significant digits. For typical use — algebra, geometry, physics homework, engineering checks — that is far more precision than needed. For very large radicands (above 2^53 ≈ 9.0 × 10^15) integer arithmetic loses precision; the calculator switches to scientific notation in that range to flag the change clearly.