Nth Root Calculator

Calculate an nth root (square root, cube root, and more) from a single number

The Nth Root Calculator is for one simple job: given a number and a whole-number index n, compute the real value x such that xⁿ equals your number. This is the everyday meaning of “nth root.” If n is 2, you are finding a square root. If n is 3, you are finding a cube root. If n is 4, you are finding a fourth root, and so on. The calculator is aimed at people who want a quick, reliable numeric answer for homework checks, spreadsheet work, basic engineering formulas, or general math tasks.

To use it, enter the number you want to take the root of and then enter the root index n as a whole number. If you want a cleaner display, you can set the number of decimal places, but it is optional and the calculator will still work if you leave it blank. After you click calculate, you will see the nth root, plus a verification check that raises the result back to the nth power. That check is important because roots often involve rounding. If the verification is extremely close to your original number, your root is correct and any small difference is just rounding or floating-point behavior.

The outputs are designed for practical use. First you get the estimated real nth root. Then you get a quick “power form” reminder that an nth root is the same as raising the number to the power of 1/n. Finally, you get a back-check that shows what happens when the answer is raised to n. If you are using the result in another calculation, this back-check gives you an immediate sanity check before you trust the value. This page intentionally stays focused on a single intent: computing a real nth root from a number and an integer n. It does not try to solve algebraic equations, complex-number roots, or multi-step exponent rules.

Assumptions and how to use this calculator

• The root index n is treated as a positive whole number. Non-integer indexes are not supported on this page.
• The calculator returns the real nth root when a real root exists. For even n and a negative input, there is no real root.
• Displayed results are rounded to your chosen decimal places (default 6). Rounding can slightly change the verification value.
• The verification step uses the displayed numeric root value, so tiny differences from the original input are normal for most non-perfect roots.
• Very large numbers or very large n can reduce precision due to typical floating-point limits in browsers.

Common questions

What does “nth root” mean in plain language?

The nth root of a number is the value that you would multiply by itself n times to get the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27. The fifth root of 32 is 2 because 2⁵ = 32. This calculator finds that value directly from your inputs and then shows a check so you can see that the relationship holds.

Why do I get an error for an even root of a negative number?

In the real number system, even roots of negative numbers do not exist. For example, there is no real number x such that x² = -9, because any real number squared is never negative. Some math contexts extend roots into complex numbers, but this calculator is intentionally limited to real roots because that matches most everyday search intent and avoids confusing outputs.

Why is the verification value not exactly the same as the number I entered?

Most roots are irrational numbers that cannot be represented exactly with a finite decimal. Browsers also use floating-point arithmetic, which stores numbers with limited precision. When the calculator rounds the root to a chosen number of decimal places, that rounded value will not raise back to the original number perfectly. The important test is closeness: if the back-check is extremely near your input, the root is correct for practical purposes.

How many decimal places should I use?

Use fewer decimals if you only need a rough estimate, for example 2 to 4 decimals for quick comparisons. Use more decimals if you plan to plug the root into another calculation and small differences matter, for example 6 to 10 decimals. If you do not know what to choose, leave it blank and use the default. The calculator also shows a verification check so you can judge whether the rounding is acceptable for your use case.

Does this calculator handle zero and one correctly?

Yes. The nth root of 0 is 0 for any positive integer n. The nth root of 1 is 1 for any positive integer n. For n = 1, the “first root” of a number is just the number itself, because x¹ = x. These are useful edge cases to remember when you are checking work or simplifying expressions.