About the Root Calculator
A root calculator finds the square root, cube root, or any nth root of a number — the value that, multiplied by itself n times, returns your input. Roots appear in geometry, standard deviation, the Pythagorean theorem, and anywhere an area or volume must be converted back to a length.
The formula
ⁿ√x = x^(1/n)x is the number under the radical, n is the degree of the root. n = 2 gives the square root, n = 3 the cube root. Roots and fractional exponents are the same operation written two ways.
How to use this calculator
- 1Enter your Number. The field starts at
64, which you can overwrite. - 2Enter your Root (e.g. 2=square, 3=cube). The field starts at
2, which you can overwrite. - 3Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Number | 64 |
| Root (e.g. 2=square, 3=cube) | 2 |
Result
√64 (Square Root): 8.000000
∛64 (Cube Root): 4.000000
2√64 (2th Root): 8.000000
Perfect Square: Yes (8)
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
A root undoes a power, which is why it is defined as a fractional exponent. Because 4³ = 64, the cube root of 64 is 4. This equivalence is practically useful: any calculator with a power function can compute any root by raising to the reciprocal power.
Even and odd roots behave differently with negative numbers. An even root of a negative number has no real value, because no real number squared is negative — that is where imaginary numbers come from. Odd roots are fine: the cube root of −8 is −2, since (−2)³ = −8.
Things worth knowing
- The square root of a negative number is not real. It is written using i, where i² = −1.
- Most roots are irrational, meaning their decimals never terminate or repeat. √2 ≈ 1.41421356 is an approximation.
- Every positive number has two square roots, one positive and one negative. The radical sign conventionally denotes the positive one.
- To take a root on any calculator, raise to the reciprocal power: the fifth root of 32 is 32^0.2 = 2.
- Roots of products separate: √(9 · 16) = √9 · √16 = 12. Roots of sums do not.
Frequently asked questions
What is a square root?+
The number that gives your input when multiplied by itself. The square root of 25 is 5 because 5 × 5 = 25. Technically −5 also qualifies, but the radical sign denotes the positive root.
Can I take the square root of a negative number?+
Not within the real numbers, because no real number squared is negative. Complex numbers handle it using i, defined so that i² = −1, giving √−9 = 3i.
How do I calculate a cube root or higher root?+
Raise the number to the reciprocal power. The cube root of 27 is 27^(1/3) = 3, and the fourth root of 81 is 81^0.25 = 3.
Why is the square root of 2 irrational?+
Because it cannot be written as a ratio of two integers — a fact proven by contradiction in ancient Greece. Its decimal expansion continues without repeating, so any written form is an approximation.