The nth root generalizes square and cube roots to any whole-number index. ⁶√1,458 asks for the number whose sixth power is 1,458; ¹⁰√1,024 asks the same with a tenth power (the answer is 2). This calculator accepts any index from 2 to 1,000, recognizes perfect powers, simplifies the radical by pulling out complete groups of factors, and gives a decimal to ten places.
How to use the nth root calculator
- Enter the index n, the small number written in the crook of the radical sign.
- Enter the radicand, the number under the radical. Decimals, fractions and negative numbers are accepted.
- Read the result on the tape: the decimal, the simplest radical form, whether the radicand is a perfect nth power, and the whole numbers the root falls between. The steps show the factorization.
Nth root formulas
| Rule | Formula |
|---|---|
| Product | ⁿ√(ab) = ⁿ√a · ⁿ√b |
| Quotient | ⁿ√(a/b) = ⁿ√a / ⁿ√b |
| Root of a root | ᵐ√(ⁿ√x) = ᵐⁿ√x |
| Simplification | ⁿ√(aⁿ · b) = a · ⁿ√b |
| Odd index, negative radicand | ⁿ√(−x) = −ⁿ√x |
Worked example: ⁶√1,458
Factor: 1,458 = 2 × 3⁶.
Group in sixes: the six 3s form one complete group, which leaves the radical as a single 3; the 2 stays inside.
Result: ⁶√1,458 = 3·⁶√2 ≈ 3 × 1.1224620483 = 3.3673861449.
Check: 3⁶ = 729 and 4⁶ = 4,096, so the root must lie between 3 and 4. ✓
Other examples: ¹⁰√1,024 = 2 exactly, ⁷√−2,187 = −3, and ¹²√4,096 = 2.
Computing roots by hand
Before calculators, nth roots were found with logarithms: log(ⁿ√x) = (log x) ÷ n. For ⁶√1,458, log₁₀ 1,458 ≈ 3.16376; dividing by 6 gives 0.52729, and 10^0.52729 ≈ 3.3674.
Newton’s method is faster when you can iterate:
Each round roughly doubles the number of correct digits, and it is essentially what this calculator does to polish the last decimal places.
Uses of nth roots
Geometric mean
The geometric mean of n positive numbers is the nth root of their product. It is the right average for rates and ratios. For growth factors 1.10, 0.95, 1.20 and 1.05 over four years, the geometric mean is ∜(1.10 × 0.95 × 1.20 × 1.05) = ∜1.3167 ≈ 1.0712, an average of about 7.1% per year. The ordinary mean of the four rates, 7.5%, would overstate the result.
Growth rates over n periods
A total growth factor F over n periods corresponds to an average rate of ⁿ√F − 1 per period. Prices that doubled over 12 years rose by ¹²√2 − 1 ≈ 5.9% per year.
Music
In equal temperament, the 12 semitones of an octave split a doubling of frequency evenly, so each semitone multiplies frequency by ¹²√2 ≈ 1.059463. That is why an A at 440 Hz is followed by a B♭ at about 466.16 Hz.
Even versus odd indexes
| Index n | ⁿ√(positive) | ⁿ√(negative) | Solutions of xⁿ = a, a > 0 |
|---|---|---|---|
| Even (2, 4, 6, …) | positive real | not real (complex) | two: ±ⁿ√a |
| Odd (3, 5, 7, …) | positive real | negative real | one |
For a coefficient in front of the radical, use the simplify radicals calculator; for fractional exponents such as x^(3/4), the exponent calculator applies the same rules.
Frequently asked questions
What is the nth root of a number?
The nth root of x is the number that gives x when raised to the nth power: ⁿ√x = y means yⁿ = x. The square root is the case n = 2 and the cube root is n = 3.
How is an nth root related to exponents?
ⁿ√x = x^(1/n). More generally, the m-th power of the nth root is x^(m/n), so 8^(2/3) = (∛8)² = 4. This is why fractional exponents and radicals are two notations for the same idea.
When does a negative number have a real nth root?
Only when n is odd. ⁷√−2,187 = −3 because (−3)⁷ = −2,187, but an even root of a negative number is not real. For even n the calculator gives the principal complex root instead.
What happens to the nth root of a number as n grows?
For any positive x, ⁿ√x approaches 1 as n increases. The 1,000th root of 2 is only about 1.000693, and the 1,000th root of 0.5 is about 0.999307.