Nth Root Calculator
Quick select root
How to use this nth root calculator
- Enter the radicand (the number you want to root) in the left field.
- Set the root index n using the quick-select tabs or by typing a custom value in the right field.
- The result appears instantly with 10 decimal places of precision.
- Read whether the number is a perfect nth power and get the complete step-by-step working above.
- Click Copy to grab the result for use in other calculations.
Nth root formula and rules
Core formula
The nth root formula is nth-root(x) = x^(1/n). Raising a number to the power 1/n gives the nth root. For x = 243 and n = 5: 243^(1/5) = 243^0.2 = 3. Verify: 3^5 = 243. This is how the calculator works for all inputs -- positive, negative (odd n only), and zero.
Generalised fractional exponent form
nth-root(x^m) = x^(m/n) -- combines a root and an exponent in one expression. Example: cube-root(x^2) = x^(2/3). This form is widely used in calculus and algebra because it converts radical expressions into standard power form, making differentiation and integration much cleaner without separate radical notation.
Product rule for nth roots
nth-root(a x b) = nth-root(a) x nth-root(b) for a, b both non-negative (or both real for odd n). This rule is essential for simplification. Example: 4th-root(48) = 4th-root(16 x 3) = 4th-root(16) x 4th-root(3) = 2 x 4th-root(3). Extract the largest perfect-nth-power factor from the radicand first.
Quick reference
| Rule | Formula | Example |
|---|---|---|
| Core formula | nth-root(x) = x^(1/n) | 5th-root(243) = 3 |
| Fractional exp. | nth-root(x^m) = x^(m/n) | cbrt(x^2) = x^(2/3) |
| Product rule | nth-root(a*b) = nth-root(a)*nth-root(b) | 4th-root(48) = 2*4th-root(3) |
| Odd negative rule | nth-root(-x) = -nth-root(x) (odd n) | 5th-root(-32) = -2 |
| Even negative | nth-root(-x) = imaginary (even n) | 4th-root(-16) undefined in R |
What is an nth root?
The nth root of x is the number y such that y^n = x. Written as x^(1/n), it generalises the square root (n=2) and cube root (n=3) to any positive integer n. The number n is called the root index or degree, and the number under the radical is the radicand. The calculator above handles all integer root indices from 2 upward.
One of the most important properties of nth roots is how the parity (odd or even) of n determines whether negative radicands are allowed. For odd indices (n = 3, 5, 7...), the nth root of a negative number is real and negative -- the 5th root of -32 is -2, because (-2)^5 = -32. For even indices (n = 2, 4, 6...), negative radicands have no real root. This is why cube roots handle negatives but square roots do not.
Nth roots appear across science, finance, and engineering wherever a quantity scales as a power of n. CAGR formulas use the nth root to find per-year rates. Acoustics, optics, and structural engineering use fractional exponents -- equivalent to nth roots -- to describe how intensity, force, or wavelength scale with physical dimensions.
- 4th-root(16) = 2 because 2^4 = 16
- 5th-root(32) = 2 because 2^5 = 32
- 6th-root(64) = 2 because 2^6 = 64
- 7th-root(-128) = -2 because (-2)^7 = -128 (odd index allows negatives)
Interpreting your nth root result
Perfect nth power -- exact integer result
When flagged as a perfect nth power, the result is an exact integer. 4th-root(81) = 3 exactly because 3^4 = 81. Perfect nth power results come up in algebraic problems, mental math, and simplification rules where an exact integer is expected or required.
Non-perfect -- irrational decimal result
Most integers are not perfect nth powers. Their roots are irrational -- the decimal expansion never ends. The calculator shows 10 decimal places. The step-by-step section verifies accuracy by raising the rounded result to the nth power and comparing it to the original radicand.
Imaginary result (even n, negative radicand)
When n is even (2, 4, 6...) and the radicand is negative, no real nth root exists. The result is imaginary. This calculator flags such inputs with a clear message and suggests switching to an odd index. For imaginary results, complex number arithmetic is required beyond this tool.
Real-world applications of nth roots
Finance -- compound annual growth rate (CAGR)
If an investment grows from $1,000 to $1,500 over 5 years, the annual growth rate is the 5th root of 1.5: 1.5^(1/5) approximately 1.0845. Subtract 1 and multiply by 100 -- approximately 8.45% per year. Every CAGR formula uses exactly this nth root calculation to convert a total multi-year return into a constant annual rate that, compounded each year, would produce that same total growth.
Engineering tolerances -- geometric mean yield
When a manufacturing process has n independent steps each contributing to total yield, the per-step rate is the nth root of the target joint yield. A circuit board with 8 process steps targeting 95% total yield requires each step to achieve at least 8th-root(0.95) approximately 99.36% yield. The nth root distributes a joint requirement equally across n independent stages.
Acoustics -- equal temperament and music theory
The 12-tone equal temperament musical scale is built on the 12th root of 2: 12th-root(2) approximately 1.05946 per semitone. Each semitone is exactly 12th-root(2) times the frequency of the previous one. After 12 semitones (one octave) the frequency doubles exactly: (12th-root(2))^12 = 2. Sound engineers and instrument makers depend on this nth root relationship for precise tuning.
Physics -- power law scaling
Many physical quantities follow power-law relationships. The radius of an n-dimensional hypersphere scales as the nth root of volume. Diffusion distances scale as the square root of time. Structural load-bearing capacity scales as a cube root or 4th root of cross-sectional area. Whenever a measurement is the nth power of a dimension, the inverse is an nth root.
Nth root reference table (radicand = 64)
Higher root indices produce results closer to 1 for the same radicand:
| Root index (n) | Expression | Result | Verification |
|---|---|---|---|
| 2nd | 64^(1/2) | 8.000000 | 8^2 = 64 |
| 3rd | 64^(1/3) | 4.000000 | 4^3 = 64 |
| 4th | 64^(1/4) | 2.828427 | 2.828427^4 approx 64 |
| 6th | 64^(1/6) | 2.000000 | 2^6 = 64 |
| 8th | 64^(1/8) | 1.681793 | 1.681793^8 approx 64 |
| 12th | 64^(1/12) | 1.414214 | sqrt(2) approx 1.414 |
How to estimate nth roots by hand
Mental estimation follows the same bracketing technique as square roots, using nth powers as brackets:
- Bracket between consecutive nth powers. For 4th-root(50): 2^4 = 16 and 3^4 = 81. So 4th-root(50) lies between 2 and 3.
- Interpolate linearly. 50 is (50-16)/(81-16) = 34/65 approximately 0.52 of the way. Estimate: 2 + 0.52 approximately 2.52. Actual: 2.659 (linear interpolation underestimates slightly for concave functions).
- Refine with one Newton step. New guess = ((n-1) x guess + x/guess^(n-1)) / n. For 4th-root(50) with 2.52: (3 x 2.52 + 50/2.52^3) / 4 approximately 2.67. Much closer to 2.659.
- Verify by raising to the nth power. 2.659^4 approximately 50.00 confirmed.
Quick nth root tips
- 4th root = square root twice: 4th-root(x) = sqrt(sqrt(x)). Handy shortcut on any basic calculator.
- Even index and sign: Confirm the radicand is non-negative if n is even -- negative inputs produce imaginary results.
- Powers of 2 to memorise: 2^10 = 1024, 2^8 = 256, 2^7 = 128, 2^6 = 64, 2^5 = 32. Recognise these for instant perfect-power identification.
- CAGR formula: annual rate = (end/start)^(1/n) - 1. The 1/n exponent IS the nth root.
- Fractional exponent on a calculator: x^y key with y = 1/n works for any n without a dedicated nth root button.
Common nth root mistakes to avoid
Computing even-index roots of negative numbers
The square root, 4th root, 6th root, and all other even-index roots of negative numbers are not real. If your calculator returns an error or NaN for a negative radicand with an even index, that is correct. Nth roots of negatives are only real when n is odd. If you encounter this, check your radicand sign or switch to an odd index like 3 or 5.
Confusing nth root with nth power
The 4th root of 16 is 2. The 4th power of 16 is 65,536. These are inverse operations: (nth-root(x))^n = x, and nth-root(x^n) = x. Mixing them up produces wildly wrong answers. Always check which direction the operation goes -- root or power -- before computing.
Rounding the index before calculating
If n is supposed to be exactly 5 but you use 4.99 by accident, the answer will differ. For CAGR calculations over a fixed period, use the exact integer n. Any deviation changes the exponent 1/n and introduces error. Double-check your root index before reading the result from this calculator.
Forgetting the negative sign for odd roots
The nth root of -x (odd n) is always -nth-root(x). Students sometimes compute the positive root and forget to apply the negative sign. 5th-root(-32) = -2, not +2. The step-by-step section above shows the sign-handling explicitly so you can follow the logic on any input.