How this calculator works
An nth root asks which number, raised to the power n, gives the radicand. The relation is a = r^n, so any one of the three quantities can be found when the other two are known. For odd indices a negative radicand is allowed because an odd power keeps the sign; an even index needs a radicand of zero or more because every even power is non-negative. The calculator returns the principal real root.
Formula
If a = r^n, then r = a^(1/n), a = r^n, and n = ln(a) / ln(r) for positive a and r with r ≠ 1. The root is real when a ≥ 0, or when a < 0 and n is odd.
Worked example
The cube root of 27 is 3 because 3^3 = 27. Reversing it, a root of 3 with index 4 gives radicand 81. Solving for the index of 64 from a root of 2 gives ln(64)/ln(2) = 6.
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Frequently asked questions
Why can an even root of a negative number not be real?
Any real number raised to an even power is zero or positive, so no real number squares or fourth-powers to a negative value. Such roots are complex numbers and lie outside this calculator.
What is the difference between a root and a radical?
They describe the same relationship. A radical is the written symbol √ together with its index; the root is the value it represents. Radical form can be exact, such as √2, while this calculator shows a decimal approximation.
Why does the index need to be a whole number?
Fractional indices are defined, but they behave like combinations of a power and a root, and the sign conventions become awkward. Restricting the index to whole numbers keeps the three-way solve unambiguous.
Is the answer always unique?
The principal real root is unique for the domain described. Even powers have a second negative root, such as -3 for the square root of 9, but the radical sign denotes the non-negative principal value.