Transparent method
The formula
Worked example
Example inputs
How to interpret the result
An even root of a negative number has no real answer, while an odd root of a negative number does: the square root of negative nine doesn't exist on the real number line, but the cube root of negative eight is simply negative two. Which rule applies depends entirely on whether the root itself is even or odd, not on the sign of the number being rooted.
Before you rely on it
What to check
Verify the result by raising it back to the root's power. A cube root that doesn't cube back to the original input signals a rounding or input error, not a formula problem.
The common error
Where people go wrong with root calculator
Expecting every root of a negative number to fail the way a square root does. Odd roots of negative numbers are perfectly ordinary real numbers; only even roots run into that wall.
Sensitivity evidence
How value (radicand) changes the result
| Value (radicand) | Result | Result squared (check) |
|---|---|---|
| 22 | 2.8 | 7.9 |
| 24 | 2.9 | 8.3 |
| 27worked example | 3 | 9 |
| 30 | 3.1 | 9.7 |
| 32 | 3.2 | 10.1 |
Every input, tested
Which input moves the result most
| Input | Tested from | To | Result at each end | Swing |
|---|---|---|---|---|
| Root degree | 2 | 4 | 5.2 to 2.3 | 2.9 (97%) |
| Value (radicand) | 24 | 30 | 2.9 to 3.1 | 0.223 (7.4%) |
Two variables at once
Result by value (radicand) and root degree
| Value (radicand) \ Root degree | 1 | 3 | 5 |
|---|---|---|---|
| 22 | 22 | 2.8 | 1.9 |
| 24 | 24 | 2.9 | 1.9 |
| 27 | 27 | 3 | 1.9 |
| 30 | 30 | 3.1 | 2 |
| 32 | 32 | 3.2 | 2 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| Value (radicand) | 27 |
| Root degree | 3 |
| Result | 3 |
| Result squared (check) | 9 |
Where the inputs come from
Enter the value, called the radicand, and the degree of the root: 2 for a square root, 3 for a cube root and so on. The calculator raises the value to the power of one over the degree. Even roots of negative numbers are not real numbers, while odd roots of negatives are, so the cube root of minus 8 is minus 2. The check output squares the result to confirm the answer.
When the result stops applying
Even roots of negative values give complex results outside the real numbers. Large degrees on small numbers approach 1, which can look surprising. Rational exponents and roots are equivalent but written differently.
What to compare it with
See the reverse operation on the exponent calculator, and logarithms on the logarithm page. The Pythagorean theorem and standard deviation calculators use square roots, and the quadratic formula calculator uses one inside the formula. The scientific notation page helps with very large radicands.
People also ask
Frequently asked questions
What is the difference between a square root and a cube root?
A square root is a number that multiplied by itself gives the value, and a cube root multiplied by itself three times gives it.
Can I take the square root of a negative number?
Not among real numbers. The result is imaginary, written with the unit i, so a real-number calculator reports no answer.
Why is the square root of 2 not a fraction?
It is irrational: its decimal expansion never ends or repeats, so no fraction of whole numbers equals it exactly.