Nth Root Calculator | Square & Cube
Calculate any nth root of a number including square root, cube root, and custom roots.
COMMON ROOTS
What Is the Nth Root Calculator | Square & Cube?
This nth root calculator computes ⁿ√x for any real number x and any non-zero root index n. Results appear in decimal (up to 15 places), scientific notation, and as a best rational approximation, plus a verification check confirming (result)ⁿ = x.
- ›Perfect root detection: exact integer roots (e.g. ∛27 = 3) are flagged automatically.
- ›Nearest perfect roots: for irrational results, shows which perfect roots bracket your number.
- ›Rational approximation: continued-fraction algorithm finds the simplest fraction within 1 ppm of the true root.
- ›Odd roots of negatives: ∛(−8) = −2 handled correctly, most basic calculators refuse negative inputs.
- ›Precision selector: 4 to 15 decimal places, chosen per calculation.
Formula
| Root | Notation | Formula | Example |
|---|---|---|---|
| Square root | √x | x^(1/2) | √25 = 5 |
| Cube root | ∛x | x^(1/3) | ∛27 = 3 |
| 4th root | ∜x | x^(1/4) | ∜81 = 3 |
| nth root | ⁿ√x | x^(1/n) | ¹⁰√1024 = 2 |
| Negative radicand (odd n) | ⁿ√(−x) | −(x^(1/n)) | ∛(−8) = −2 |
| Fractional exponent | xᵐ/ⁿ | (ⁿ√x)ᵐ | 8^(2/3) = 4 |
| Verification | Check | (ⁿ√x)ⁿ = x | (∛27)³ = 27 |
How to Use
- 1Enter the number x (can be negative for odd n).
- 2Enter the root index n, any non-zero real number, including decimals.
- 3Select the decimal precision from the dropdown (4–15 places).
- 4Click Calculate or press Enter on any field.
- 5Results show decimal, scientific, and fraction approximation.
- 6Click Copy to copy the decimal result to clipboard.
- 7Use the preset buttons (n=2,3,4,5,10,12,100) for instant common-root lookups.
Example Calculation
Example, Cube root of 125
Example, 5th root of −32
Example, 12th root of 2 (music theory)
Fractional exponents xᵐ/ⁿ
Understanding Nth Root | Square & Cube
Understanding the nth Root
The nth root of x is the value y satisfying yⁿ = x. It generalises the square root (n=2) and cube root (n=3) to any index. The formula ⁿ√x = x^(1/n) follows from exponent laws: (x^(1/n))ⁿ = x^(n/n) = x. All exponent identities apply: ⁿ√(ab) = ⁿ√a × ⁿ√b, and ⁿ√(a/b) = ⁿ√a / ⁿ√b.
- ›Perfect nth roots: integers where the root is also an integer (27 is a perfect cube, 81 a perfect 4th power).
- ›Irrational roots: most nth roots are non-terminating, non-repeating decimals.
- ›Negative radicands: allowed for odd n only, (−2)³ = −8, so ∛(−8) = −2.
- ›Non-integer n: e.g. x^(1/2.5) = x^0.4, valid for any non-zero real n.
Negative Numbers and Odd Roots
An even root of a negative number is not real, it is complex. √(−4) = 2i. This calculator returns a clear informative message for these cases. An odd root of a negative number is always real and negative: ∛(−8) = −2, ⁵√(−32) = −2, ⁷√(−128) = −2. This is because an odd power of a negative number is negative: (−2)³ = −8. Most basic calculators fail silently for negative inputs to any root, this calculator handles them correctly.
Practical Applications of nth Roots
The nth root appears across science, engineering, music, and finance:
- ›Music: ¹²√2 ≈ 1.05946, the equal-temperament semitone ratio (12 steps × this = one octave = 2× frequency).
- ›Finance: ¹²√(1 + annual_rate) − 1 = monthly equivalent rate for compound interest.
- ›Physics: r = ∛(3V / 4π), sphere radius from volume.
- ›Statistics: geometric mean of n values = ⁿ√(x₁ × x₂ × … × xₙ).
- ›Economics: ¹²√(CPI_end / CPI_start) − 1 = average annual inflation rate over 12 years.
Rational Approximations via Continued Fractions
Every irrational root can be closely approximated by simple fractions. The continued fraction algorithm finds the best rational approximation p/q: the fraction closest to the true value among all fractions with denominator ≤ q. For √2, the sequence is 1/1 → 3/2 → 7/5 → 17/12 → 41/29 → 99/70, each fraction better than any other with a smaller denominator. These approximations are used in gear ratio engineering (to achieve near-irrational speed ratios with integer tooth counts) and in musical tuning, the fraction 18/17 ≈ 1.0588 was used in Baroque lute tuning as a close rational approximation to the exact ¹²√2.
Frequently Asked Questions
What is the nth root and how does it differ from a square root?
The square root is the 2nd root (n=2). The nth root generalises to any index: ⁿ√x is the value y with yⁿ = x.
- ›Square root (n=2): √25 = 5, because 5² = 25
- ›Cube root (n=3): ∛27 = 3, because 3³ = 27
- ›4th root (n=4): ∜81 = 3, because 3⁴ = 81
- ›General: ⁿ√x = x^(1/n) by exponent law
Square roots arise in 2D geometry (Pythagorean theorem, distance formula). Cube roots arise in volume problems. Higher roots appear in compound-growth calculations, signal power, and spectral analysis.
Can I take the nth root of a negative number?
It depends on whether n is odd or even:
- ›Odd n (1, 3, 5, …): real result, negative. ∛(−8) = −2 because (−2)³ = −8.
- ›Even n (2, 4, 6, …): not real. √(−4) = 2i, a complex number.
- ›This calculator handles odd roots correctly and gives a clear message for even roots of negatives.
In complex arithmetic, every non-zero number has exactly n nth roots distributed around the unit circle in the complex plane.
What is a perfect nth root?
A perfect nth root is a number whose nth root is exactly an integer. Perfect squares: 1, 4, 9, 16, 25, …. Perfect cubes: 1, 8, 27, 64, 125, …. Perfect 4th powers: 1, 16, 81, 256, 625, ….
This calculator detects exact integer roots and flags them. For non-perfect roots, it shows the nearest perfect roots on either side for context.
How do I compute fractional exponents like x^(2/3)?
A fractional exponent xᵐ/ⁿ means: take the nth root, then raise to the mth power: xᵐ/ⁿ = (ⁿ√x)ᵐ.
To use this calculator: set n to the reciprocal of m/n. For x^(2/3), set n = 3/2 = 1.5, giving x^(1/1.5) = x^(2/3) ✓.
Why is the 12th root of 2 important in music theory?
In equal temperament (standard Western tuning), an octave spans 12 semitones and doubles the frequency. Each semitone must multiply frequency by ¹²√2 ≈ 1.05946, so that 12 steps produce exactly 2× the frequency.
This irrational ratio replaced historical tuning systems (Pythagorean, meantone) that used rational ratios for perfect intervals but couldn't modulate freely between all 12 keys.
How does the rational fraction approximation work?
The continued fraction algorithm finds the best rational approximation p/q to the root. Every real number x has a continued fraction: x = a₀ + 1/(a₁ + 1/(a₂ + …)). Truncating at each level gives convergents, increasingly accurate fractions, each the best possible for its denominator size.
These approximations are used in gear ratio design, historical musical tuning (18/17 for semitone), and antenna element spacing in antenna arrays.
Can n be a decimal or negative?
Yes, any real non-zero value of n is supported:
- ›n = 2.5: x^(1/2.5) = x^0.4, used in power-law scaling (metabolic rate ∝ mass^0.75 uses 4/3 root).
- ›n = −2: x^(1/−2) = 1/√x, the reciprocal of the square root.
- ›n = 1: x^(1/1) = x, trivially the number itself.
- ›n = 0.5: x^(1/0.5) = x², raises to the square instead of extracting a root.