Square Root Calculator
Find the square root, cube root, and nth root of any number. Instantly checks if the number is a perfect square or perfect cube.
How to Use
Enter a number to get its square root (ā), cube root (ā), and squared/cubed values. To find an nth root, also enter the root index (e.g., 4 for the fourth root).
The Extended Calculator adds a visual root diagram, tabs for cube roots and nth roots, and a perfect squares reference table. The Professional Calculator shows Newton's Method convergence steps, complex roots of quadratics, and radical simplification.
Root Formulas
Square Root: āx = x^(1/2) ā the number that, multiplied by itself, gives x
Cube Root: āx = x^(1/3) ā the number that, cubed, gives x
Nth Root: āæāx = x^(1/n)
Perfect Square: x is a perfect square if āx is a whole number
Newton's Method: xāāā = (xā + S/xā) / 2 (for āS)
Worked Example
ā225: 225 = 15 Ć 15 ā 15 (perfect square)
ā27: 27 = 3 Ć 3 Ć 3 ā 3 (perfect cube)
4th root of 81: 81^(1/4) = 3
ā72 simplified: 72 = 36 Ć 2 ā ā72 = 6ā2
| Iteration | Approximation | Error |
|---|---|---|
| 0 | 7 | 7.1068e-2 |
| 1 | 7.07142857 | 3.6076e-4 |
| 2 | 7.07106782 | 9.2023e-9 |
| 3 | 7.07106781 | 0.0000e+0 |
| 4 | 7.07106781 | 0.0000e+0 |
| 5 | 7.07106781 | 0.0000e+0 |
| 6 | 7.07106781 | 0.0000e+0 |
| 7 | 7.07106781 | 0.0000e+0 |
| 8 | 7.07106781 | 0.0000e+0 |