The formula
How it works
A root is the opposite of a power. The square root of a number is what you multiply by itself to get it; the cube root uses three copies, and so on. Enter a number and the index to find any root — with support for cube and higher roots of negative numbers.
FAQ
Can you take the root of a negative number?
Only for odd roots. The cube root of −8 is −2, because (−2)³ = −8. Even roots, like the square root, have no real answer for negatives, so the calculator shows them as undefined.
How is a root related to a power?
Taking the nth root is the same as raising to the power 1/n. So the square root is the ½ power and the cube root is the ⅓ power — which is exactly how the calculator computes them.
Why does the square root symbol not show an index?
By convention, a bare radical sign \(\sqrt{x}\) always means the square root, so the index 2 is left out. Every other root, like the cube root \(\sqrt[3]{x}\), must show its index.
What is the principal root?
When a number has more than one valid root, the principal root is the positive one the calculator returns. For example 16 has two square roots, 4 and −4, but the principal root shown is 4.
Can you find the root of a decimal or fraction?
Yes — the same formula works for any positive number. The square root of 0.25 is 0.5, and the square root of 2.25 is 1.5, since the calculator just raises the number to the power 1/n.
Is the result always a whole number?
No — most roots are irrational and go on forever without repeating, like the square root of 2. The calculator rounds the decimal result so it fits on screen, but the true value is not exact unless the number is a perfect power.
What happens with a non-integer index?
The formula still applies: it raises the number to 1/n even when n has a decimal, such as 2.5. Non-integer indices are less common but useful for modelling growth rates that compound at fractional exponents.
About the root calculator
This calculator finds the nth root of a number — the square root, cube root, or any higher root you need. A root answers the question “what number, multiplied by itself this many times, gives my number?” Roots are the reverse of exponents, and they appear whenever you need to undo a power, from working out the side of a square to the growth rate hidden inside compound interest.
How to use it
Type the number you want the root of, then the index — 2 for a square root, 3 for a cube root, and so on. The calculator returns the result instantly. For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27, and the square root of 16 is 4. You can take roots of negative numbers when the index is odd: the cube root of −27 is −3. Even roots of negatives have no real value and are shown as undefined.
The formula
The nth root is written , and it is the same as raising to a fractional power, . That connection is why it undoes an exponent: since , taking the nth root and then the nth power gets you back to the start. The square root is the special case , usually written without the little number, because it is by far the most common.
Where it is used
Roots turn up across maths, science and finance. Geometry uses square roots in the Pythagorean theorem and to find the side of a square from its area, while the cube root gives the edge of a cube from its volume. Statistics uses the square root to turn variance into standard deviation, physics uses it in formulas for time and speed, and finance uses roots to find average growth rates over several years.