Root Calculator
The calculator finds square roots, cube roots and roots of any index, simplifies radicals and shows a step-by-step solution.
The root calculator finds the square root, the cube root and a root of any index from 2 to 100. For whole numbers it simplifies the radical (for example, √72 = 6√2) and shows the approximate value.
What is an n-th root
The n-th root of a number a is the number that gives a when raised to the power n:
n√a = b, if bn = a
The root of index 2 is called the square root, and the index is not written: √9 = 3, because 32 = 9. The root of index 3 is the cube root: 3√8 = 2, because 23 = 8.
An even root of a negative number does not exist among real numbers. An odd root of a negative number is negative: 3√−27 = −3.
How to simplify a radical
- Break the radicand into prime factors.
- Group equal factors into groups of n — each group comes out of the root as a single factor.
- The remaining factors stay under the root.
Example 1. Simplify √72.
72 = 23 · 32 = (2 · 3)2 · 2 = 62 · 2
√72 = √(62 · 2) = 6√2 ≈ 8.485
Example 2. Find 3√216.
216 = 23 · 33 = 63
3√216 = 6
Square roots worth remembering
| a | 1 | 4 | 9 | 16 | 25 | 36 |
|---|---|---|---|---|---|---|
| √a | 1 | 2 | 3 | 4 | 5 | 6 |
| a | 49 | 64 | 81 | 100 | 121 | 144 |
|---|---|---|---|---|---|---|
| √a | 7 | 8 | 9 | 10 | 11 | 12 |
If the root is not a whole number, the calculator shows its value to 10 decimal places and checks it by raising it to the power.