Radicals and Roots Calculator

Find the nth root of any real number, positive or negative. This tool gives you the principal (real) root along with the complete set of roots, including the complex ones most calculators skip. Working with exponents instead? Try our Percentage Decrease Calculator, Percentage Difference Calculator, or Quadratic Formula Calculator.

Enter Your Values

Quick Reference: Common Roots

Expression Meaning Result
∛81 (n=4)4th root of 81±3
∛8 (n=3)Cube root of 82
∛-27 (n=3)Cube root of -27-3
∛1024 (n=5)5th root of 10244

Result

Principal, real root:

All roots:

Showing the first 20 roots. Contact us if you need the rest.

Share this calculation:

Enter n and x, then hit Calculate to see the roots here.

How to Use This Calculator

  1. Type the root you want into the n field. For a square root that's 2, for a cube root it's 3, and so on n can be any whole number.
  2. Type the number you're taking the root of into the x field. This can be positive or negative.
  3. Click Calculate. You'll get the principal real root right away, plus the full list of roots underneath it.
  4. Use Copy to grab the answer as plain text, or Print if you need a paper copy without all the page clutter.
  5. Share your exact calculation with someone else using the link that appears it'll open with the same numbers already filled in.

Practical Example / Formula Used

Taking the nth root of a number x is the same as raising x to the power of 1/n. That's really all a radical is under the hood:

∛x (n) = x1/n

Here's a worked example. Say n = 25 and x = 18. Plugging that into the formula gives 181/25, which works out to roughly 1.12256346. That's the principal root the single real number you'd normally think of as "the answer."

But here's the part a lot of calculators leave out: every nth root actually has n total solutions once you allow complex numbers. For n = 25, that means there are 25 numbers that, when raised to the 25th power, all give you back 18. Most of them just aren't real numbers they involve i, the imaginary unit. This calculator lists those out too, using the standard De Moivre's formula approach: each root sits at an equal angle around a circle in the complex plane, all sharing the same distance from zero (that distance being |x|1/n).

Why Use This Calculator?

  • It handles negative bases correctly, including the tricky case where an even root of a negative number has no real answer at all.
  • You get every root, not just the principal one useful if you're studying complex numbers or checking work for a class.
  • The share link means you can send someone the exact calculation instead of retyping numbers over text or email.
  • Everything runs in your browser, so there's no waiting on a server round-trip for something this quick.

Key Factors / Variables Explained

n - the index of the root, i.e. which root you're taking. n = 2 is a square root, n = 3 is a cube root, and higher values work the same way. It has to be a positive whole number.

x - the number you're finding the root of, sometimes called the radicand. It can be positive, negative, or zero.

Principal root - the one real-number answer that's returned when you use a calculator's or programming language's built-in root/power function. For a positive x, it's always positive. For a negative x, it only exists when n is odd.

Complex roots - the remaining n − 1 solutions that involve the imaginary unit i. They come in conjugate pairs (a + bi and a − bi) except for the principal root itself.

Frequently Asked Questions

Because no real number, multiplied by itself an even number of times, ever produces a negative result. Try squaring −3 you get 9, not −9. The same logic applies to any even root, so those cases only have complex solutions.

They're basically the same idea. "Radical" usually refers to the symbol and the expression written with it, while "root" refers to the value itself. When people say "the 5th root of 32," they mean the same thing as writing 32 under a radical with a small 5 tucked into the notch.

Once you bring complex numbers into the picture, an nth root equation technically has n solutions, not just one. Most everyday math only cares about the single real answer (the principal root), which is why it's the one shown first here.

Not on this page - n needs to be a whole number here since it represents which root you're taking. If you're working with a fractional exponent like x2/3, our fractional exponents calculator is the better fit.

Not in terms of raw size every root of a given number has the exact same distance from zero on the complex plane. The principal root is just the one that lands on the real number line, which is why it's treated as "the" answer in everyday use.

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