Root Calculator
Calculate the square root, cube root, or any nth root of a number.
- Original Value27
- 3√ Result3
Sizing up 27 against its 3-degree root — the further apart the bars, the more root-taking shrinks the number.
ƒShow your work
- 13√27 = 3
- 2Check: 3^3 ≈ 27
About the Root Calculator
A root is the inverse of an exponent — instead of asking 'what do you get when you raise this number to a power,' it asks 'what number, raised to this power, gives you the value you started with.' The square root is the familiar one, but cube roots and higher-degree roots follow the exact same idea and show up whenever a problem runs an exponent in reverse: solving for a side length from a known area or volume, or checking algebra homework that involves un-squaring or un-cubing an expression.
This calculator finds the nth root of any value for any degree n, including roots that don't come out to a clean whole number — which is most of them, since perfect squares and perfect cubes are the exception rather than the rule.
How it’s calculated
The nth root of a value is the number that, multiplied by itself n times, produces that value. It's computed as value^(1/n) — raising to a fractional exponent is mathematically identical to taking a root, which is why a root calculator and a fractional-exponent calculator are really doing the same operation.
Even roots (square root, 4th root, and so on) of a negative number aren't real numbers — no real number multiplied by itself an even number of times can produce a negative result, since a negative times a negative is always positive. Odd roots (cube root, 5th root) of a negative number are fine and simply come out negative, since a negative multiplied by itself an odd number of times stays negative.
Frequently asked questions
What's the difference between a square root and a cube root?
A square root asks what number squared (multiplied by itself twice) gives the value; a cube root asks what number cubed (multiplied by itself three times) gives it. √9 = 3 because 3² = 9; ∛27 = 3 because 3³ = 27.
Can you take an even root of a negative number?
Not as a real number — no real number squared, or raised to any even power, can produce a negative result. Even roots of negative numbers only exist in the complex number system, which is outside what this calculator handles.
How are roots and fractional exponents related?
They're the same operation written two ways: the nth root of a value equals that value raised to the power of 1/n. A cube root is the same as raising to the power 1/3, and there's no mathematical difference between the two notations.
How can I estimate a square root without a calculator?
Find the two perfect squares your number sits between, then interpolate. √50 sits between √49 (=7) and √64 (=8), and since 50 is close to 49, the root should be a bit above 7 — about 7.07, which is in fact the real answer.
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