Nth Root Calculator
Calculate the square root, cube root or any nth root of a number, including real roots of negative numbers for odd degrees.
3
- Simplest radical form
- ∛27 = 3
- As a power
- 27^(1/3)
- Scientific notation
- 3 × 10⁰
Nearby roots
The same root of the whole numbers around x. Press a number to work it out.
| 2.80203933066 | |
| 2.84386697985 | |
| 2.88449914061 | |
| 2.92401773821 | |
| 2.96249606841 | |
| 3 | |
| 3.03658897188 | |
| 3.07231682569 | |
| 3.10723250595 | |
| 3.14138065239 | |
| 3.17480210394 |
The working — Newton’s method
Start from an estimate b₀ above the root (here, the answer rounded up to one significant figure), then repeat b = [b × (n − 1) + x / bⁿ⁻¹] / n until it stops changing.
Compare with another root
Rules for radicals
- Product
- ⁿ√(ab) = ⁿ√a · ⁿ√b
- Quotient
- ⁿ√(a / b) = ⁿ√a / ⁿ√b
- Power of a root
- (ⁿ√a)ᵐ = ⁿ√(aᵐ) = a^(m/n)
- Root of a root
- ᵐ√(ⁿ√a) = ᵐⁿ√a
- As a power
- ⁿ√a = a^(1/n)
- Rationalizing a denominator
- a / √b = a√b / b
- Adding — like radicals only
- p·ⁿ√a + q·ⁿ√a = (p + q)·ⁿ√a
- Multiplying different degrees, l = lcm(m, n)
- ᵐ√a · ⁿ√b = ˡ√(a^(l/m) · b^(l/n))
For an even degree, a and b are taken to be zero or more.
What this nth root calculator does
Enter a number and a root degree n, and this tool instantly returns the real nth root. Set n to 2 for a square root, 3 for a cube root, or any other value for a higher root — there are quick buttons for the two most common cases. The result updates live as you type, and a Copy button puts the answer on your clipboard. It handles decimals and large numbers as well as whole ones.
How roots of negative numbers work
The calculator is sign-aware. A negative number has a real root only when the degree is an odd whole number: for example the cube root of −8 is −2, because (−2) × (−2) × (−2) = −8. Even roots — like the square root — of a negative number are not defined over the real numbers, so the tool tells you no real root exists instead of returning a misleading value — and gives the complex answer under it, 2i and −2i for the square root of −4, or the principal root a + bi for a higher even degree. A positive number has two real roots of any even degree, and an "Also a root" row gives the negative one beside the principal, positive one. Internally it computes the root as the magnitude raised to the power 1/n and then restores the original sign for odd degrees. Asked for more decimal places than a double can hold, it works the digits out exactly instead — for a whole degree and a number of moderate length, and says so when it cannot — and a "Rules for radicals" fold lists the product, quotient, power and root-of-a-root laws with rationalizing and adding like radicals. Folded under the answer, “Nearby roots” gives the same root of the whole numbers around x, each a button that works it out, and “Compare with another root” sets a second root beside the first and says which is larger and by what ratio; “Reset” puts the page back where it opened.
Frequently asked questions
How do I get a square root or cube root?
Set the root degree n to 2 for a square root or 3 for a cube root — the quick buttons do this for you. Any other whole or decimal n gives the corresponding higher root.
Why does the cube root of a negative number work but the square root does not?
A negative number has a real root only for odd degrees. The cube root of −8 is −2 because multiplying −2 by itself three times gives −8, but no real number squared gives a negative result, so an even root of a negative number is undefined over the reals.
Does it simplify the root, or show how it was worked out?
Yes. The answer also comes in simplest radical form, as a power with a fractional exponent, and in scientific notation. The working shows Newton's method step by step, and you choose how many decimal places the answer is rounded to.
Can it give more digits than a calculator, or the complex root of a negative number?
Yes. Past the places a double can hold, "Decimal places" works the root out exactly from the digits you typed, for a whole degree and a number of moderate length, and says when it cannot. Under "No real root" the complex answer is given — 2i and −2i for the square root of −4 — and a positive number’s second square root appears as "Also a root".
Can I compare two roots, or step to the roots of nearby numbers?
Yes. “Compare with another root” takes a second number and its degree, gives both values and says which is larger and by what ratio, or that they are equal, and says in words when the second has no real root. “Nearby roots” lists the same root of the whole numbers either side of x, each a button that sets x to it.