Nth Root Calculator
Find the nth root of any real number: enter the number and the root degree (2 for square root, 3 for cube root, or any degree up to 100). The calculator returns the exact decimal value instantly.
Nth Root Solver
Nth Root Formula
Where:
- \(x\) — the number (radicand) you want to take the root of
- \(n\) — the degree of the root (a positive integer: 2 for a square root, 3 for a cube root, and so on)
- \(\sqrt[n]{x}\) — the nth root of \(x\): the number \(y\) that satisfies \(y^n = x\)
The formula expresses the nth root as a fractional exponent. Taking the nth root of a number is exactly the same as raising that number to the power \(1/n\), so the two notations are interchangeable.
Nth Root – Calculation Example
Example: find the cube root of 8 (the calculator's default values: Number = 8, Degree of the root = 3).
\[ \sqrt[3]{8} = 8^{\frac{1}{3}} = 2 \]
Check: \(2^3 = 2 \times 2 \times 2 = 8\), so the answer is correct.
Result: the cube root of 8 is 2.
How to Use the Nth Root Calculator
- Type the number you want to extract the root from into the Number field (for example, 8).
- Enter the root degree in the Degree of the root field — use 2 for a square root, 3 for a cube root, or any integer from 1 to 100.
- Read the result in the Nth Root field. The answer is rounded to 6 decimal places.
To verify a result, raise it back to the power \(n\) with the multiplication calculator — you should get your original number back.
What Is an Nth Root?
The nth root generalizes the familiar square root and cube root to any degree. The nth root of a number \(x\) is the value \(y\) that, when multiplied by itself \(n\) times, gives \(x\). In symbols: \(y = \sqrt[n]{x}\) means \(y^n = x\). The square root (\(n=2\)) and cube root (\(n=3\)) are simply the two most common special cases.
The key idea behind the formula is the equivalence between roots and fractional exponents. Because \(\sqrt[n]{x} = x^{1/n}\), any root operation can be rewritten as exponentiation. This makes roots easy to evaluate with a calculator and convenient to combine with other exponent rules, for example \(\sqrt[n]{x^m} = x^{m/n}\).
Nth roots appear everywhere in practice. In geometry, the side length of a cube with volume \(V\) is \(\sqrt[3]{V}\), and the side of a square with area \(A\) is \(\sqrt{A}\). In finance, the compound annual growth rate is computed as an nth root: over \(n\) years, the average growth factor is \(\sqrt[n]{\text{final}/\text{initial}}\). In statistics, the geometric mean of \(n\) values is the nth root of their product, and the quadratic mean (root mean square) finishes with a square root.
When interpreting the result, remember that for a positive radicand and an odd degree there is exactly one real root, and for an even degree the principal (positive) root is the standard answer. For example, both \(2\) and \(-2\) squared give 4, but \(\sqrt{4} = 2\) by convention.
Common Roots Reference
| Degree \(n\) | Name | Example |
|---|---|---|
| 2 | Square root | \(\sqrt{25} = 5\) |
| 3 | Cube root | \(\sqrt[3]{27} = 3\) |
| 4 | Fourth root | \(\sqrt[4]{16} = 2\) |
| 5 | Fifth root | \(\sqrt[5]{32} = 2\) |
Assumptions, Limitations and Tips
- Degree range: the calculator accepts root degrees from 1 to 100. A degree of 1 returns the number itself, since \(x^1 = x\).
- Negative numbers: odd roots of negative numbers are well defined (for example, \(\sqrt[3]{-8} = -2\)), and the calculator handles them correctly. Even roots of negative numbers, however, have no real result — \(\sqrt{-4}\) is not a real number — so use negative inputs only with odd degrees.
- Rounding: results are rounded to 6 decimal places. Many roots, such as \(\sqrt{2} \approx 1.414214\), are irrational, so the displayed value is an approximation.
- Non-integer degrees: the calculator expects an integer degree. For expressions like \(x^{2.5}\), rewrite them using exponent rules (e.g. \(x^{2.5} = x^2 \sqrt{x}\)) before calculating.
- Sanity check: after any calculation, raise the result to the power \(n\) — you should recover the original number (up to rounding).
Frequently Asked Questions
How do I calculate an nth root by hand without a calculator?
The two main methods are prime factorization and estimation. For factorization, break the number into primes: \(\sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3} = 2 \times 3 = 6\). This works cleanly only for perfect powers. Otherwise, estimate: pick a guess, raise it to the nth power, and adjust up or down until you converge. For square roots, the long-division-like digit-by-digit algorithm also works.
Why does the calculator show an approximate result like 1.414214?
Most roots are irrational numbers — their decimal expansions never terminate or repeat. \(\sqrt{2}\), \(\sqrt[3]{5}\), and \(\sqrt[4]{10}\) cannot be written exactly as finite decimals or fractions. The calculator rounds the true value to 6 decimal places, so raising the displayed result back to the nth power recovers your original number only up to a tiny rounding error.
What happens with a negative number under an even root?
An even root of a negative number has no real solution. There is no real number \(y\) with \(y^2 = -4\), because any real number squared is non-negative. The answers exist only in the complex numbers (e.g. \(\sqrt{-4} = 2i\)). Odd roots of negative numbers, however, are perfectly real: \(\sqrt[3]{-27} = -3\) and \(\sqrt[5]{-32} = -2\).
Why does an even root of a positive number have two possible answers?
Both \(2\) and \(-2\) satisfy \(y^2 = 4\), and both \(3\) and \(-3\) satisfy \(y^4 = 81\). By mathematical convention, the radical symbol \(\sqrt[n]{x}\) always denotes the principal (non-negative) root. The calculator follows this convention. If you need all real solutions to \(y^n = x\) with even \(n\), remember the negative counterpart.
Can I use a decimal or fractional degree, like the 2.5th root?
This calculator only accepts integer degrees from 1 to 100. For a decimal exponent, rewrite it using exponent rules. For example, \(x^{2.5} = x^{5/2} = x^2\sqrt{x}\): square \(x\), then take the square root. Similarly, \(x^{0.5} = \sqrt{x}\) is just a square root with degree 2, and \(x^{1/3}\) is a cube root with degree 3.
What is the most common mistake when computing nth roots?
A frequent error is mixing up the root degree with the exponent: \(\sqrt[n]{x}\) means raising to the power \(1/n\), not to the power \(n\). For example, \(\sqrt[3]{8} = 8^{1/3} = 2\), while \(8^3 = 512\). Another mistake is forgetting the sign: entering \(-32\) with degree 4 gives no real answer, but with degree 5 it correctly returns \(-2\).
How are nth roots used to average growth rates or find geometric means?
The geometric mean of \(n\) positive numbers is the nth root of their product: \(\sqrt[n]{x_1 x_2 \cdots x_n}\). The compound annual growth rate works the same way: if an investment grows from \(P\) to \(F\) over \(n\) years, the annual growth factor is \(\sqrt[n]{F/P}\), and the CAGR is that factor minus 1. Note that CAGR assumes smooth compounding and does not reflect year-to-year volatility or fees.