How to use
- Enter the number. Negative numbers and decimals work.
- Enter the root: 2 for a square root, 3 for a cube root, or any whole number up to 99.
- Read the decimal value, the simplest radical form and the step-by-step working.
How the square root calculator works
A square root of x is a number that gives x when multiplied by itself. The √ sign means the principal root, the one that is not negative.
To simplify a root, find the largest perfect-square factor (or perfect cube for ∛), take its root outside the sign and leave the rest inside. In the UK, a root left in this form is called a surd.
- Odd roots of negative numbers are negative: ∛−125 = −5.
- Square roots of negative numbers are imaginary: √−12 = 2√3 i.
- Even roots of 4 or more of a negative number are not real numbers.
Examples
- √72: 72 = 36 × 2, so √72 = 6√2 ≈ 8.485281374.
- √500: 500 = 100 × 5, so √500 = 10√5 ≈ 22.36067977.
- ∛16: 16 = 8 × 2, so ∛16 = 2∛2 ≈ 2.5198421.
- √60: it lies between 7 and 8, because 7² = 49 and 8² = 64. Exactly, √60 = 2√15 ≈ 7.745966692.
Squares, cubes and square roots
| n | n² | n³ | √n |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 1.4142 |
| 3 | 9 | 27 | 1.7321 |
| 4 | 16 | 64 | 2 |
| 5 | 25 | 125 | 2.2361 |
| 6 | 36 | 216 | 2.4495 |
| 7 | 49 | 343 | 2.6458 |
| 8 | 64 | 512 | 2.8284 |
| 9 | 81 | 729 | 3 |
| 10 | 100 | 1000 | 3.1623 |
| 11 | 121 | 1331 | 3.3166 |
| 12 | 144 | 1728 | 3.4641 |
Exact answers, rounding and complex roots
Simplest radical form is shown for whole numbers up to 1,000,000,000,000. The decimal follows your decimal-places setting; exact roots such as ∛64 = 4 are shown without rounding errors.
Some advanced texts define (−8)^(1/3) as a complex number. This calculator follows the algebra-course rule that odd roots of negative numbers are real, so ∛−8 = −2.
To raise a number to a power instead, including fractional powers like 2/3, use the exponent calculator.