Square, cube and nth root calculator
Calculate roots of any index, including negative numbers with an odd index.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Calculate roots of any index, including negative numbers with an odd index.
- Formula used
ⁿ√a = a^(1/n)- Example
- Square root of 81:
Root calculator
A root is the inverse of a power: if 3⁴ = 81, then the fourth root of 81 is 3.
Leave the index at 2 for a square root, set it to 3 for a cube root, or use any other value, decimals included.
How it works
Internally the root is computed as a power with a fractional exponent. For negative numbers a real solution exists only when the index is odd, because an even product of negatives is always positive.
Formula
ⁿ√a = a^(1/n)
(ⁿ√a)ⁿ = a
If a < 0, the real root exists only when n is odd
Worked example
Square root of 81:
81^(1/2) = 9
Cube root of −27:
(−27)^(1/3) = −3
Check: (−3)³ = −27 ✓
Explanation
Why the root of a negative does not exist in the reals
No real number multiplied by itself gives a negative result: positive times positive is positive, and negative times negative is too. That is why the square root of minus one is not among the reals. Mathematicians defined it as the imaginary unit, and from it come the complex numbers, indispensable in electrical engineering and quantum physics.
Cube roots do accept negatives
Unlike square roots, the cube root of minus eight is minus two, because minus two cubed is minus eight. The general rule is that odd indices accept negative radicands and even ones do not.
Roots are fractional powers
The square root of a number is that number to the power of one half, and the cube root to the power of one third. Seeing it that way unifies both operations and explains why they combine: the fourth root is the square root of the square root.
Rationalising denominators
School mathematics asks you to remove roots from the denominator by multiplying top and bottom by the same root. The habit dates from hand calculation, when dividing by a long decimal was more awkward than multiplying by it.
Frequently asked questions
What is the square root of a negative number?
It does not exist among the real numbers. It exists among the complex numbers, but this calculator works with reals.
Can I use a decimal index?
Yes. An index of 2.5 is the same as raising to the power 1/2.5 = 0.4.
Can I take the cube root of a negative number?
Yes. Odd indices accept negative radicands: the cube root of minus eight is minus two.
What is a root written as a power?
An nth root is equivalent to raising to one over n. The square root is the power one half and the cube root the power one third.
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