Calculate square roots, cube roots, and any nth root of a number. See all common roots in a single table.
n = 2 is square root (√), n = 3 is cube root (∛), n = 4 is fourth root, etc.
| Root | Notation | Result | Verification |
|---|---|---|---|
| Square | √64 | 8PERFECT | 8^2 ≈ 64 |
| Cube | ∛64 | 4PERFECT | 4^3 ≈ 64 |
| 4th | 4√64 | 2.82842712 | 2.82842712^4 ≈ 64 |
| 5th | 5√64 | 2.29739671 | 2.29739671^5 ≈ 64 |
| 6th | 6√64 | 2PERFECT | 2^6 ≈ 64 |
| 8th | 8√64 | 1.68179283 | 1.68179283^8 ≈ 64 |
| 10th | 10√64 | 1.51571657 | 1.51571657^10 ≈ 64 |
Perfect root (shown in green) means the result is an exact integer. e.g. √64 = 8 exactly.
You have a square garden bed with 90 square feet of planting area and need to know how long to cut each side board. That is a square root, and the answer, 9.4868 feet or about 9 feet 5.8 inches, is not something you can guess to the inch. Enter the area, leave the root at 2, and the page returns the side length together with the 2nd through 10th roots of the same number so you can compare them.
Take 90 and ask for the 2nd, 3rd and 4th roots. The page shows 9.4868329805, 4.4814047466 and 3.0800702882. Raise the first of those back to the power 2 and you land on 90 again, which is what the Verification column in the table does, rounded to four decimals.
The formula behind each row is the nth root of x = x^(1/n). For n = 2 that is 90^0.5, for n = 3 it is 90 raised to one third, computed with JavaScript's Math.pow. One third cannot be stored exactly in binary, so tiny errors creep in: the cube root of 1000 is held internally as 9.999999999999998. The page rounds to 10 decimal places (8 in the table), so you see a clean 10 and a green PERFECT tag, which is awarded when the result is within 1e-9 of a whole number.
Signs follow the odd and even rule. An odd root keeps the sign, so the cube root of -27 is -3 (the page takes the root of 27 and puts the minus back). An even root of a negative number has no real value, and the page answers with the message 'Complex result' instead of a number. The root of 0 is 0 for any n.
The result box is rounded to 10 decimals and the table to 8. Roots such as the square root of 2 never end, so the verification column will be within rounding of your number, not identical to it.
Not here. The page flags it as complex and shows no value. By hand, the square root of -9 is 3i, but this calculator does not produce imaginary numbers.
No, and mixing them up is common. 27 ÷ 3 is 9, while the cube root of 27 is 3, because 3 × 3 × 3 = 27. The root undoes a power; division splits into equal parts.
Bracket it with perfect squares. 9² is 81 and 10² is 100, so the square root of 90 sits between 9 and 10, and since 9.5² is 90.25 it should be just under 9.5.